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By Yiu P.

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Extra resources for Notes on Euclidean geometry

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C X P Q A B 2. Let P be a point on the circumcircle of triangle ABC, with orthocenter H. The midpoint of P H lies on the nine-point circle of the triangle. 5 5 YIU: Euclidean Geometry 40 3. (a) Let ABC be an isosceles triangle with a = 2 and b = c = 9. Show that there is a circle with center I tangent to each of the excircles of triangle ABC. (b) Suppose there is a circle with center I tangent externally to each of the excircles. Show that the triangle is equilateral. (c) Suppose there is a circle with center I tangent internally to each of the excircles.

YIU: Euclidean Geometry 55 To construct the two circles tangent to the minor arc, the chord AB, and the circle (KP ), we proceed as follows. (1) Let C be the midpoint of the major arc AB. Complete the rectangle BM CD, and mark on the line AB points A0 , B 0 such that A0 M = M B 0 = M D. (2) Let the perpendicular to AB through P intersect the circle (O) at P1 and P2 . A0 (3) Let the circle passing through P1 , P2 , and 0 intersect the chord B Q AB at 0 . Q P1 A' A Q' M P B Q B' O C P2 D Then the circles tangent to the minor arc and to the chord AB at Q and Q0 are also tangent to the circle (KP ).

Suppose the smaller semicircles have radii a and b respectively. Let Q be the intersection of the largest semicircle with the perpendicular through P to AB. This perpendicular is an internal common tangent of the smaller semicircles. Q H U V a A a-b b O1 O K b P O2 B A O1 O P O2 Exercise 1. Show that the area of the shoemaker’s knife is πab. 2. Let U V be the external common tangent of the smaller semicircles. Show that U P QV is a rectangle. 3. Show that the circle through U , P , Q, V has the same area as the shoemaker’s knife.

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Notes on Euclidean geometry by Yiu P.

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