Math 621: Elementary Geometry |
1. If \(t_a = AL\) is the bisctor of angle \(A\) in triangle
\(\Delta ABC\), show that \[\dfrac{\bar{BL}}{\bar{LC}} =
\dfrac{AB}{AC}.\]
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
\(\bar{AD}\cdot\bar{BC} + \bar{BD}\cdot\bar{CA} + \bar{CD}\cdot\bar{AB} = 0.\)
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.
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 \(\ell\) at a point \(A\) on \(\ell\), where \(A\) 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.) \(\Sigma_1\) and \(\Sigma_2\) intersect; 2.)
\(\Sigma_1\) is inside of \(\Sigma_2\); 3.) \(\Sigma_1\) is outside of
\(\Sigma_2\).
13. Let \(\Sigma_1\), \(\Sigma_2\), and \(\Sigma_3\) be non-concentric,
non-intersecting ordinary circles. Describe the locus of all points \(P\) for
which \(P^{\Sigma_1} = P^{\Sigma_2} = P^{\Sigma_3}\) in case: 1.) the centers
\(O_1, O_2, O_3\) are collinear; 2.) the centers are not collinear. Do the
construction for both cases.
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