Math 621: Elementary Geometry

Midterm Exam: Review Guide

Both Chapters 1 and 2:

You must be able to state any definitions and named theorems from chapters 1 and 2.

You must be able to sketch pictures (not constructions) to illustrate the major named theorems: Ceva, Menelaus, Desargues, etc.

Chapter 1:

1. You must be able to complete the following constructions:

	•   bisect an angle
	•   divide a segment into \(n\) equal pieces
	•   add, subtract, multiply, divide two numbers
	•   construct a square root
	•   construct the tangent line to a circle through a point.

2. You must be able to construct a triangle given:

	•   \(a\), \(b\), \(c\)
	•   \(A\), \(b\), \(c\)
	•   \(A\), \(B\), \(c\)
	•   \(A\), \(c\), \(h_c\)
	•   \(b\), \(c\), \(m_c\)
	•   \(B\), \(h_c\), \(b + c\)
	•   \(A\), \(B\), \(b + c\)

3. Prove the Pythagorean Theorem: If \(\Delta ABC\) is a right triangle with right angle \(C\), then \(a^2 + b^2 = c^2\).

Chapter 2:

1. If \(t_a = AL\) is the bisctor of angle \(A\) in triangle \(\Delta ABC\), show that \[\dfrac{\overline{BL}}{\overline{LC}} = \dfrac{AB}{AC}.\]
2. Prove that the medians of a triangle are concurrent.

3. Let \(\Sigma\) be an excircle of \(\Delta ABC\); that is, \(\Sigma\) is a circle outside of \(\Delta ABC\) that is tangent to the three (extended) sides of the triangle at points \(A', B', C'\), where \(A'\) lies on \(BC\), \(B'\) lies on \(AC\), and \(C'\) lies on \(AB\). Prove that \(AA'\), \(BB'\), and \(CC'\) are concurrent. Hint: duplicate the proof of the Gergonne point, with appropriate changes.

4. Prove Euler's theorem: If \(A, B, C, D\) are collinear, then \(\overline{AD}\cdot\overline{BC} + \overline{BD}\cdot\overline{CA} + \overline{CD}\cdot\overline{AB} = 0.\)

5. Use Euler's theorem to show that if \((AB,CD) = r\), then \((AC,BD) = 1-r\).

6. Construct the harmonic conjugate of \(C\) with respect to \(AB\) in three different ways, when \(C\) is either inside or outside of \(AB\).

7. Let \(AB\) be a line segment with midpoint \(M\), and let \(P\) be any point not collinear with \(AB\). Use the fact that the harmonic conjugate of \(M\) is an ideal point to contruct a line through \(P\) parallel to \(AB\).

8. Consider a triangle \(\Delta ABC\) with harmonic conjugate pairs \((P,P')\) of (extended) side \(BC\), \((Q,Q')\) of (extended) side \(AC\), and \((R,R')\) of (extended) side \(AB\). Prove that \(P, Q, R\) are collinear if and only if \(AP'\), \(BQ'\), and \(CR'\) are concurrent. Hint: recall that \((P,P')\) is a harmonic conjugate pair of \(BC\) if and only if \((BC,PP') = -1\), etc.

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