10/9/2017 Elementary Geometry

http://www.math.wichita.edu/~ryan/teaching/M621/midterm/exam_1_study.html 1/2

Math 621: Elementary Geometry

Midterm Exams

Chapter 2: Strongly Recommended Exercises

This page contains exercises that should closely approximate problems

on the Chapter 2 exam. You should also review all written homework

assignments from Chapter 2.

1. If is the bisctor of angle in triangle , show that

2. Prove that the medians of a triangle are concurrent.

3. Let be an excircle of ; that is, is a circle outside of that is tangent

to the three (extended) sides of the triangle at points , where lies on ,

lies on , and lies on . Prove that , , and are concurrent. Hint:

duplicate the proof of the Gergonne point, with appropriate changes.

4. Prove Euler’s theorem: If are collinear, then

5. Use Euler’s theorem to show that if , then .

6. Construct the harmonic conjugate of with respect to in three different ways,

when is either inside or outside of .

7. Let be a line segment with midpoint , and let be any point not collinear with

. Use the fact that the harmonic conjugate of is an ideal point to contruct a line

through parallel to .

8. Consider a triangle with harmonic conjugate pairs of (extended) side

, of (extended) side , and of (extended) side . Prove that

are collinear if and only if , , and are concurrent. Hint: recall that

is a harmonic conjugate pair of if and only if , etc.

t

a = AL A ΔABC

= .

B¯L

L¯C

AB

AC

Σ ΔABC Σ ΔABC

A′, B′, C ′ A′ BC

B′ AC C ′ AB AA′ BB′ CC ′

A, B, C, D

A¯D ⋅ B¯C + B¯D ⋅ C¯A + C¯D ⋅ A¯B = 0.

(AB, CD) = r (AC, BD) = 1 – r

C AB

C AB

AB M P

AB M

P AB

ΔABC (P, P ′)

BC (Q, Q′) AC (R, R′) AB

P, Q, R AP ′ BQ′ CR′

(P, P ′) BC (BC, PP ′) = -1

10/9/2017 Elementary Geometry

http://www.math.wichita.edu/~ryan/teaching/M621/midterm/exam_1_study.html 2/2

9. Construct a circle orthogonal to a given circle and passing through two points not on

the circle. Make sure you can do: 1.) both points inside; 2.) one inside and one outside;

3.) both points outside.

10. Construct a circle orthogonal to a given circle and tangent to a given line at a point

on , where is not a point on the circle.

11. Construct a circle orthogonal to two given circles and passing through a given point

not on either circle.

12. Construct the radical axis of any two non-concentric ordinary circles. Be sure you can

do all cases: 1.) and intersect; 2.) is inside of ; 3.) is outside of .

13. Let , , and be non-concentric, non-intersecting ordinary circles. Describe the

locus of all points for which in case: 1.) the centers

are collinear; 2.) the centers are not collinear. Do the construction for both cases.

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ℓ

A ℓ A

Σ1 Σ2 Σ1 Σ2 Σ1 Σ2

Σ1 Σ2 Σ3

P P Σ1 = P Σ2 = P Σ3 O1, O2, O3

## Elementary Geometry

10
Oct