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Illustration used to prove the corollary that "From a point outside a line there exists only one perpendicular to the line."

Perpendicular to Line Corollary

Illustration used to prove the corollary that "From a point outside a line there exists only one perpendicular…

Illustration of a triangle with interior segments and angles labeled.

Segments and Angles in a Triangle

Illustration of a triangle with interior segments and angles labeled.

Illustration of a triangle with interior segments and angles labeled.

Segments and Angles in a Triangle

Illustration of a triangle with interior segments and angles labeled.

Illustration used to prove that "If two sides of a triangle are unequal, the angle opposite the greater side is greater than the angle opposite the less side."

Sides of Triangle Theorem

Illustration used to prove that "If two sides of a triangle are unequal, the angle opposite the greater…

Illustration used to prove that "The sum of any two sides of a triangle is greater than the third side."

Sides of Triangle Theorem

Illustration used to prove that "The sum of any two sides of a triangle is greater than the third side."

Illustration used to prove that "If two triangles have two sides of one equal respectively to two sides of the other, but the included angle of the first greater than the included angle of the second, then the third side of the first is greater than the third side of the second."

2 Triangles Theorem

Illustration used to prove that "If two triangles have two sides of one equal respectively to two sides…

Illustration of triangle ABC with BE extended through the triangle at point D. Segment AB is equal to segment BD.

Segments Labeled In A Triangle

Illustration of triangle ABC with BE extended through the triangle at point D. Segment AB is equal to…

Illustration of a quadrilateral with diagonals AC and BD.

Quadrilateral With Diagonals

Illustration of a quadrilateral with diagonals AC and BD.

Illustration used to prove that "If two triangles have two sides of one equal respectively to two sides of the other, but the third side of the first greater than the third side of the second, then the angle opposite the third side of the first is greater than the angle opposite the third side of the second."

2 Triangles Theorem

Illustration used to prove that "If two triangles have two sides of one equal respectively to two sides…

Illustration of the construction used to create an isosceles triangle, given the bases and the sum of the altitude and a side.

Construction Of An Isosceles Triangle

Illustration of the construction used to create an isosceles triangle, given the bases and the sum of…

Illustration used to prove that "If two straight lines are parallel to a third straight line, they are parallel to each other."

Parallel Lines Theorem

Illustration used to prove that "If two straight lines are parallel to a third straight line, they are…

Illustration of two straight lines cut by a transversal. The 8 angles formed are labeled.

2 Lines Cut By A Transversal

Illustration of two straight lines cut by a transversal. The 8 angles formed are labeled.

Illustration used to prove the theorem, "If two straight lines are cut by a transversal making a pair of alternate interior angles equal, the lines are parallel."

Parallel Lines Cut By A Transversal Theorem

Illustration used to prove the theorem, "If two straight lines are cut by a transversal making a pair…

Illustration of two straight lines, m and n, cut by a transversal t.

2 Lines Cut By A Transversal

Illustration of two straight lines, m and n, cut by a transversal t.

Illustration of two straight lines, AB and CD, cut by a transversal EF.

2 Lines Cut By A Transversal

Illustration of two straight lines, AB and CD, cut by a transversal EF.

Illustration of the construction used to create a line parallel to a given line.

Construction Of A Parallel Line

Illustration of the construction used to create a line parallel to a given line.

Illustration used to prove the theorem, "If two parallel lines are cut by a transversal, the alternate interior angles are equal."

Parallel Lines Cut By A Transversal Theorem

Illustration used to prove the theorem, "If two parallel lines are cut by a transversal, the alternate…

Illustration used to prove the corollary that "Two lines perpendicular respectively to two intersecting lines also intersect."

Intersecting Lines Corollary

Illustration used to prove the corollary that "Two lines perpendicular respectively to two intersecting…

Illustration used to prove the theorem, "Two angles whose sides are parallel, each to each, are either equal or supplementary."

2 Angles With Parallel Sides Theorem

Illustration used to prove the theorem, "Two angles whose sides are parallel, each to each, are either…

Illustration used to prove the theorem, "Two angles whose sides are perpendicular, each to each, are either equal or supplementary."

2 Angles With Perpendicular Sides Theorem

Illustration used to prove the theorem, "Two angles whose sides are perpendicular, each to each, are…

Illustration used to prove the theorem, "The sum of the angles of any triangle is two right angles."

Sum Of Angles In Triangle Theorem

Illustration used to prove the theorem, "The sum of the angles of any triangle is two right angles."

A circle with labels for radius, diameter, and circumference. The visual will help to remember what each term means.

Circle Parts

A circle with labels for radius, diameter, and circumference. The visual will help to remember what…

"In the triangle above, the line AB is its altitude. Since we know how to find the area of one triangle, we can find the areas of as many triangles as we have made from our circle. Therefore, to find the area of a circle: Find the area of one of the triangles and multiply by the number of triangles." -Foster, 1921

Area of Circle with Triangles

"In the triangle above, the line AB is its altitude. Since we know how to find the area of one triangle,…

An illustration showing the construction used to divide a line AB into two equal parts; and to erect a perpendicular through the middle. "With the end A and B as centers, draw the dotted circle arcs with a radius greater than half the line. Through the crossings of the arcs draw the perpendicular CD, which divides the line into two equal parts."

Construction Of A Line Divided In Equal Parts

An illustration showing the construction used to divide a line AB into two equal parts; and to erect…

An illustration showing the construction used to erect a perpendicular. "With C as a center, draw the dotted circle arcs at A and B equal distances from C. With A and B as centers, draw the dotted circle arcs at D. From the crossing D draw the required perpendicular DC."

Construction Of A Perpendicular

An illustration showing the construction used to erect a perpendicular. "With C as a center, draw the…

An illustration showing the construction used to erect a perpendicular from a point to a line. "With C as a center, draw the dotted circle arc so that it cuts the line at A and B. With A and B as centers, draw the dotted cross arcs at D with equal radii. Draw the required perpendicular through C and crossing D."

Construction Of A Perpendicular

An illustration showing the construction used to erect a perpendicular from a point to a line. "With…

An illustration showing the construction used to erect a perpendicular at the end of a line. "With the point D as a center at a distance from the line, and with AD as radius, draw the dotted circle arc so that it cuts the line at E through E and D, draw the diameter EC: then join C and A, which will be the required perpendicular."

Construction Of A Perpendicular

An illustration showing the construction used to erect a perpendicular at the end of a line. "With the…

An illustration showing the construction used to erect a parallel line. "With C as a center, draw the dotted arc ED, with E as a center, draw through C the dotted arc F.C. With the radius FC and E as a center, draw the cross arc at D. Join C with the cross at D, which will be the required parallel line.

Construction Of A Parallel

An illustration showing the construction used to erect a parallel line. "With C as a center, draw the…

An illustration showing the construction used to erect an equal angle. "With D as a center, draw the dotted arc CE: and with the same radius and B as a center, draw the arc GF; then make GF equal to CE; then join BF, which will form the required angle, FBG=CDE."

Construction Of An Equal Angle

An illustration showing the construction used to erect an equal angle. "With D as a center, draw the…

An illustration showing the construction used to divide an angle into two equal parts. "With C as a center, draw the dotted arc DE; with D and E as centers, draw the cross arcs at F with equal radii. Join CF, which divides the angle into the required parts."

Construction Of A Divided Angle

An illustration showing the construction used to divide an angle into two equal parts. "With C as a…

An illustration showing the construction used to divide an angle into two equal parts when the lines do not extend to a meeting point. "Draw the lined CD and CE parallel, and at equal distances from the lines AB and FG. With C as a center, draw the dotted arc BG; and with B and G as centers, draw the cross arcs H. Join CD, which divides the angle into the required equal parts."

Construction Of A Divided Angle

An illustration showing the construction used to divide an angle into two equal parts when the lines…

An illustration showing the construction used to erect a parallelogram given two sides and an angle. "Draw the base line DE, and make the angle FDE = C; lines DE = B and DF = A; complete the parallelogram by cross arcs at G, and the problem is thus solved."

Construction Of A Parallelogram

An illustration showing the construction used to erect a parallelogram given two sides and an angle.…

An illustration showing the construction used to divide the line AB in the same proportion of parts as AC. "Join C and B, and through the given divisions 1, 2, and 3 draw lines parallel with CB, which solves the problem."

Divide A Line Proportionately

An illustration showing the construction used to divide the line AB in the same proportion of parts…

An illustration showing how to find the center of a circle which will pass through three given points A, B, and C. "With B as a center, draw the arc DEFG; and with the same radius and A as a center, draw the cross arcs D and F; also with C as a center, draw the cross arcs E and G. Join D and F, and also E and G, and the crossing o is the required center of the circle."

Find The Center Of A Circle Through 3 Points

An illustration showing how to find the center of a circle which will pass through three given points…

An illustration showing how to construct a square upon a given line. "With AB as radius and A and B as centers, draw the circle arcs AED and BEC. Divide the arc BE in two equal parts at F, and with EF as radius and E as center, draw the circle CFD. Join A and CB and D, C and D, which completes the required square."

Square Constructed Upon A Given Line

An illustration showing how to construct a square upon a given line. "With AB as radius and A and B…

An illustration showing how to construct a tangent to a circle through a given point in a circumference. "Through a given point A and center C, draw the line BC. With A as a center, draw the circle arcs B and C; with B and C as centers, draw the cross arcs D and E; then join D and E, which is the required tangent."

Construction Of Tangent To Circle

An illustration showing how to construct a tangent to a circle through a given point in a circumference.…

An illustration showing how to construct a tangent to a circle through a given point outside of a circumference. "Join A and C, and upon AC as a diameter draw the half circle ABC, which cuts the given circle at B. Join A and B, which is the required tangent."

Construction Of Tangent To Circle

An illustration showing how to construct a tangent to a circle through a given point outside of a circumference.…

An illustration showing how to construct a tangent circle to a circle with a given radius. "Through the given point C, draw the diameter AC extended beyond D: from C set off the given radius R to D; then D is the center of the required circle, which tangents the given circle at C."

Construction Of Circle Tangent To Circle

An illustration showing how to construct a tangent circle to a circle with a given radius. "Through…

An illustration showing how to construct a tangent circle to 2 given circles. "Join centers C and c of the given circles, and extend the line to D; draw the radii AC and ac parallel with one another. Join Aa, and extend the line to D. On CD as a diameter, draw the half circle CeD; on cD as a diameter, draw the half circle cfD; then the crossings e and f are tangenting points of the circles."

Construction Of Circle Tangent To 2 Circles

An illustration showing how to construct a tangent circle to 2 given circles. "Join centers C and c…

An illustration showing how to construct a tangent to 2 given circles of different diameters. "Join the centers C and c of the given circles, and extend the line to D; draw the radii AC and ac parallel with one another. Join Aa, and extend the line to D. On CD as a diameter, draw the half circle CeD; on cD as a diameter, draw the half circle cfD; then the crossings e and f are the tangenting points of the circles."

Construction Of Tangent To 2 Circles

An illustration showing how to construct a tangent to 2 given circles of different diameters. "Join…

An illustration showing how to construct a tangent between 2 given circles. "Join the centers C and c of the given circles; draw the dotted circle arcs, and join the crossing m, n, which line cuts the center line at a. With aC as diameter, draw the half circle afC; and with ac as a diameter, draw the half circle cea; then the crossings e and f are the tangenting points of the circles."

Construction Of Tangent Between 2 Circles

An illustration showing how to construct a tangent between 2 given circles. "Join the centers C and…

An illustration showing how to construct a circle tangent to a given line and given circle. "Add the given radius r to the radius R of the circle, and draw the arc cd. Draw the line ce parallel with and at a distance r from the line AB. Then the crossing c is the center of the required circle that will tangent the given line and circle."

Construction Of A Circle Tangent To A Line And A Circle

An illustration showing how to construct a circle tangent to a given line and given circle. "Add the…

An illustration showing how to construct the center and radius of a circle that will tangent a given circle. "Through the given point C, draw the tangent GF; bisect the angle FGE; then o is the center of the required circle that will tangent AB at C, and the line DE."

Construction Of A Center And Radius Of A Circle That Will Tangent A Given Circle

An illustration showing how to construct the center and radius of a circle that will tangent a given…

An illustration showing how to construct the center and radius of a circle that will tangent a given circle and line. "Through the given point C, draw the line EF at right angles to AB; set off from C the radius r of the given circle. Join G and F. With G and F as centers draw the arc crosses m and n. Join mn, and where it crosses the line EF is the center of the required circle."

Construction Of A Center And Radius Of A Circle That Will Tangent A Given Circle And Line

An illustration showing how to construct the center and radius of a circle that will tangent a given…

An illustration showing how to construct the center and radius of a circle that will tangent a given circle and line. "From C, erect the perpendicular CG; set off the given radius r from C to H. With H as a center and r as radius, draw the cross arcs on the circle. Through the cross arcs draw the line IG; then G is the center of the circle arc FIC, which tangents the line at C and the circle at F."

Construction Of A Center And Radius Of A Circle That Will Tangent A Given Circle And Line

An illustration showing how to construct the center and radius of a circle that will tangent a given…

An illustration showing how to construct two circles that tangent themselves and two given lines. "Draw the center line AB between the given lines; assume D to be the tangenting point of the circles; draw DC at right angles to AB. With C as center and CD as radius, draw the circle EDF. From E, draw Em at right angles to EF; and from F draw Fm at right angles to FE; then m and n are the centers for the required circles."

Construction Of Two Circles That Tangent Themselves and 2 Given Lines

An illustration showing how to construct two circles that tangent themselves and two given lines. "Draw…

An illustration showing how to construct a circle that tangents two given lines inclined to one another with the one tangenting point being given. "Draw the center line GF. From E, draw EF at right angles to AB; then F is the center of the circle required.

Construction Of A Circle That Tangents 2 Given Lines

An illustration showing how to construct a circle that tangents two given lines inclined to one another…

An illustration showing how to construct a circle that tangents two given lines and goes through a given point c on the line FC, which bisects the angle of the lines. "Through C draw AB at right angles to CF; bisect the angles DAB and EBA, and the crossing on CF is the center of the required circle."

Construction Of A Circle That Tangents 2 Given Lines And Goes Through A Given Point

An illustration showing how to construct a circle that tangents two given lines and goes through a given…

An illustration showing how to construct a cyma, or two circle arcs that will tangent themselves, and two parallel lines at given points A and B. "Join A and B; divide AB into four equal parts and erect perpendiculars. Draw Am at right angles from A, and Bn at right angles from B; then m and n are the centers of the circle arcs of the required cyma."

Construction Of A Cyma

An illustration showing how to construct a cyma, or two circle arcs that will tangent themselves, and…

An illustration showing how to construct a talon, or two circle arcs that will tangent themselves, and meet two parallel lines at right angles in the given points A and B. "Join A and B; divide AB into four equal parts erect perpendiculars; then m and n are the centers of the circle arcs of the required talon."

Construction Of A Talon

An illustration showing how to construct a talon, or two circle arcs that will tangent themselves, and…

An illustration showing how to construct a circle arc without recourse to its center, but its chord AB and height h being given. "With the chord as radius, and A and B as centers, draw the dotted circle arcs AC and BD. Through the point O draw the lines AOo and BOo. Make the arcs Co=Ao and Do=Bo. Divide these arcs into any desired number of equal parts, and number them as shown on the illustration. Join A and B with the divisions, and the crossings of equal numbers are points in the circle arc."

Construction Of A Circle Arc

An illustration showing how to construct a circle arc without recourse to its center, but its chord…

An illustration showing how to construct a center and radius of a circle that will tangent the three sides of a triangle. "Bisect two of the angles in the triangle, and the crossing C is the center of the required circle."

Construction Of The Center And Radius Of A Circle Tangent To Triangle Sides

An illustration showing how to construct a center and radius of a circle that will tangent the three…

An illustration showing how to construct an equilateral triangle inscribed in a circle. "With the radius of the circle and center C draw the arc DFE; with the same radius, and D and E as centers, set off the points A and B. Join A and B, B and C, C and A, which will be the required triangle."

Construction Of An Equilateral Triangle Inscribed In A Circle

An illustration showing how to construct an equilateral triangle inscribed in a circle. "With the radius…

An illustration showing how to construct a square inscribed in a circle. "Draw the diameter AB, and through the center erect the perpendicular CD, and complete the square as shown in the illustration."

Construction Of A Square Inscribed In A Circle

An illustration showing how to construct a square inscribed in a circle. "Draw the diameter AB, and…

An illustration showing how to construct a square circumscribed about a circle. "Draw the diameters AB and CD at right angles to one another; with the radius of the circle, and A, B, C, and D as centers, draw the four dotted half circles which cross one another in the corners of the square, and thus complete the problem."

Construction Of A Square Circumscribed About A Circle

An illustration showing how to construct a square circumscribed about a circle. "Draw the diameters…

An illustration showing how to construct a pentagon inscribed in a circle. "Draw the diameter AB, and from the center C erect the perpendicular CD. Bisect the radius AC at E; with E as center, and DE as radius, draw the arc DE, and the straight line DF is the length of the side of the pentagon."

Construction Of A Pentagon Inscribed In A Circle

An illustration showing how to construct a pentagon inscribed in a circle. "Draw the diameter AB, and…

An illustration showing how to construct a pentagon on a given line. "From B erect BC perpendicular to and half the length of AB; join A and C prolonged to D; with C as center and CB as radius, draw the arc BD; then the chord BB is the radius of the circle circumscribing the pentagon. With A and B as centers, and BD as radius, draw the cross O in the center."

Construction Of A Pentagon On A Line

An illustration showing how to construct a pentagon on a given line. "From B erect BC perpendicular…

An illustration showing how to construct a pentagon on a given line without resort to its center. "From B erect Bo perpendicular to and equal to AB; with C as center and Co as radius, draw the arc Do, then AD is the diagonal of the pentagon. With AD as radius and A as center, draw the arc DE; and with E as center and AB as radius, finish the cross E, and thus complete the pentagon."

Construction Of A Pentagon On A Line

An illustration showing how to construct a pentagon on a given line without resort to its center. "From…

An illustration showing how to construct a hexagon in a given circle. "The radius of the circle is equal to the side of the hexagon."

Construction Of A Hexagon In A Circle

An illustration showing how to construct a hexagon in a given circle. "The radius of the circle is equal…

An illustration showing how to construct a heptagon, or septagon. "The appotem a in a hexagon is the length of the side of the heptagon. Set off AB equal to the radius of the circle; draw a from the center C at right angles to AB; then a is the required side of the heptagon."

Construction Of A Heptagon

An illustration showing how to construct a heptagon, or septagon. "The appotem a in a hexagon is the…