## Elementary Geometry

10 Oct

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
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 ABC
B
AC C AB AABBCC
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
BQCR
(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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A ℓ A
Σ
1 Σ2 Σ1 Σ2 Σ1 Σ2
Σ1 Σ2 Σ3
P P Σ1 = P Σ2 = P Σ3 O1, O2, O3