Proposition XLVII

170. Draw any triangle and lines perpendicular to its sides, bisecting them. Do these lines intersect in a point? How do the distances of the point from the vertices of the triangle compare?

Theorem. The perpendicular bisectors of the sides of a triangle pass through a point which is equidistant from the vertices of the triangle.

Data: Any triangle, as ABC, and FG, DK, and EJ, the perpendicular bisectors of AB, BC, and AC respectively.

To prove that FG, DK, and EJ pass through a point which is equidistant from A, B, and C.

E

K

A

F

B

Proof. Since two of the perpendiculars will intersect, if sufficiently produced (why ?), it needs to be shown only that the third perpendicular passes through the point of intersection, to prove that the three pass through the same point.

Let any two of the perpendiculars, as FG and DK, intersect

at H.

Then, § 103, H is equidistant from A and B, and also from B and C.

Hence,

H is equidistant from 4 and C;

.. § 104, H lies in the perpendicular bisector of AC.

That is, EJ passes through the point H, which is equidistant from A, B, and C.

Therefore, etc.

Q.E.D.

Ex. 116. The middle points of the sides of an equilateral triangle are joined. What kind of triangles are formed?

Ex. 117. How do the lines drawn from the middle points of the equal sides of an isosceles triangle to the opposite extremities of the base compare in length?

Ex. 118. The parallel sides of a trapezoid are 35 and 55 feet respectively. What is the length of the line joining the middle points of the non-parallel sides?

Ex. 119. If from the extremities of the shorter base of an isosceles trapezoid perpendiculars are drawn to the longer base, two triangles are formed. How do they compare?

Proposition XLVIII

171. Draw any triangle and lines from the vertices perpendicular to the opposite sides. Do these lines intersect in a point?

Theorem. The perpendiculars from the vertices of a triangle to the opposite sides pass through the same point.

Proof. Through the vertices A, B, and C draw GH, GK, and HK parallel to BC, AC, and AB respectively, and intersecting in G, H, and K.

Hence, § 170, AD, BE, and CF pass through the same point. Therefore, etc.

Q.E.D.

Ex. 120. How many sides has a polygon the sum of whose exterior angles is double the sum of its interior angles?

Ex. 121. How many sides has a polygon the sum of whose interior angles is equal to the sum of its exterior angles?

Ex. 122. The perimeter of an isosceles triangle is 176 feet, and the base is 1 times one of the equal sides. What is the length of each side of the triangle ?

Ex. 123. How many sides has an equiangular polygon which can be divided into equilateral triangles by lines drawn from a point within to the vertices of the polygon?

SUMMARY

172. Truths established in Book I.·

1. Two lines are equal,

a. If they can be made to coincide.

b. If they are sides of an equilateral triangle.

to any point in the perpendicular erected at its middle point.

e. If they are sides of a triangle opposite equal angles.

j. If they are the sides of a square.

i. If they are the non-parallel sides of an isosceles trapezoid.

h. If they represent the distances of any point in the bisector of an angle from its sides.

g. If they are drawn from any point in a perpendicular to a line and cut off equal distances on that line from the foot of the perpendicular.

c. If they represent the distances from the extremities of a straight line

d. If they are homologous sides of equal triangles.

f. If they are sides of an equiangular triangle.

§ 36

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§ 103

§ 108

§ 118

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$ 134

§ 139

k. If they are the sides of a rhombus.

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1. If they are parallel and are intercepted between parallel lines. m. If they are opposite sides of a parallelogram.

§ 144

§ 151

p. If one joins the middle points of the non-parallel sides of a trapezoid and the other is equal to half the sum of the parallel sides.

o. If one is half a side of a triangle and the other is drawn parallel to it and bisecting one of the other sides.

n. If they are parts intercepted on one transversal by parallel lines which intercept equal parts on another transversal.

§ 153

§ 157

§ 158

§ 160

2. Two lines are parallel,

a. If both are perpendicular to the same line.

b. If when cut by a transversal the alternate interior angles are equal.

871

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c. If when cut by a transversal the corresponding angles are equal. § 77 d. If when cut by a transversal the sum of the two interior angles on the same side of the transversal is equal to two right angles.

e. If both are parallel to a third line.

f. If they are the bases of a trapezoid.

g. If they are opposite sides of a parallelogram.

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§ 140

h. If one is a side of a triangle and the other joins the middle points of the other two sides.

i. If one is either base of a trapezoid and the other joins the middle points of the non-parallel sides.

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§ 160

3. Two lines are perpendicular to each other,

a. If they form one or more right angles with each other.

b. If one is perpendicular to a line which is parallel to the other.

§ 26

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PLANE GEOMETRY. - BOOK I.

y two or more points in one are each equidistant from the extre other.

§§ 106,

e is the base of an isosceles triangle and the other is the bise ical angle.

lines form one and the same straight line,

ey are the sides of a straight angle.

ey are the exterior sides of adjacent supplementary angles.

o lines are unequal,

S

ne is a perpendicular from a point to a straight line and the of er line from that point to the straight line.

hey represent the distances from the extremities of a straight int without the perpendicular erected at its middle point. hey are sides of a triangle and lie opposite unequal angles. hey are the third sides of two triangles whose other sides are eq ach, and include unequal angles.

hey are drawn from any point in a perpendicular to a line and al distances on that line from the foot of the perpendicular. hey are distances cut off on a line from the foot of a perpendic nequal lines from any point in the perpendicular.

one is any side of a triangle and the other is equal to the sum of o sides. §§ 124,

one is equal to the sum of two lines from a point within a tria tremities of one side, and the other is equal to the sum of the of

it is the base of an isosceles triangle, by the bisector of the vert S it is the base of an isosceles triangle, by a perpendicular from § it is either diagonal of a parallelogram, by the other diagonal. § it is the side of a triangle, by a straight line drawn parallel to d bisecting the other side.

ines pass through the same point,

they are the medians of a triangle.

f they are the bisectors of the three angles of a triangle.

f they are the perpendicular bisectors of the sides of a triangle. §

f they are perpendiculars from the vertices of a triangle to the of

g.

If they are alternate interior angles formed by a transversal and parallel lines.

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h. If they are corresponding angles formed by a transversal and parallel lines.

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i. If their sides are parallel and both pairs extend in the same or in opposite directions from their vertices.

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j. If their sides are perpendicular to each other and both angles are acute or both are obtuse.

k. If they are angles of an equiangular triangle.

9. If they are angles of an equilateral triangle.

p. If they are opposite the equal sides of an isosceles triangle.

o. If they are the third angles of two triangles whose other angles are equal, each to each.

m. If they are formed by the perpendicular bisector of a straight line and lines from any point in it to the extremities of the straight line. n. If they are homologous angles of equal triangles.

§ 83 § 98 7. If they are formed adjacent to a straight line by lines joining the extremities of that line with any point in its perpendicular bisector.

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§ 105

§ 108

§ 113

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r. If they are the opposite angles of a parallelogram.

§ 117

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10. Two angles are supplementary,

a. If their sum is equal to two right angles.

c. If their corresponding sides are perpendicular and one angle is acute and the other obtuse.

b. If their corresponding sides are parallel and one pair extends in the same direction and the other in opposite directions from their vertices.

§ 32

§ 81

§ 83

11. Two angles are unequal,

a. If they are angles of a triangle and lie opposite unequal sides. b. If they are the angles opposite unequal sides of two triangles whose other two sides are equal, each to each.

§ 126

§ 130

12. An angle is bisected,

b. If it is the vertical angle of an isosceles triangle, by a line from the vertex perpendicular to the base. § 122

a. If it is the vertical angle of an isosceles triangle, by the perpendicular bisector of the base.

§ 121

angle.

c. By a line every point of which is equidistant from the sides of the § 135