# Notes on euclidean geometry by Kiran Kedlaya

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2. Given circles C1 , C2 , C3 , C4 such that Ci and Ci+1 are externally tangent for i = 1, 2, 3, 4 (where C5 = C1 ). Prove that the four points of tangency are concyclic. 3. (Romania, 1997) Let ABC be a triangle, D a point on side BC and ω the circumcircle of ABC. Show that the circles tangent to ω, AD, BD and to ω, AD, DC are tangent to each other if and only if ∠BAD = ∠CAD. 4. (Russia, 1995) Given a semicircle with diameter AB and center O and a line which intersects the semicircle at C and D and line AB at M (M B < M A, M D < M C).

DIAGRAM What happens to the point O? Points near O get sent very far away, in all different directions, so there is no good place to put O itself. To rectify this, we define the inversive plane as the usual plane with one additional point, called the point at infinity. ) We extend inversion to the entire inversion plane by declaring that O and ∞ are inverses of each other. As an aside, we note a natural interpretation of the inversive plane. Under stereographic projection (used in some maps), the surface of a sphere, minus the North Pole, is mapped to a plane tangent to the sphere at the South Pole as follows: a point on the sphere maps to the point on the plane collinear with the given point and the North Pole.

IMO 1994 proposal) The incircle of ABC touches BC, CA, AB at D, E, F , respectively. X is a point inside ABC such that the incircle of XBC touches BC at D also, and touches CX and XB at Y and Z, respectively. Prove that EF ZY is a cyclic quadrilateral. 44 8. (Israel, 1995) Let P Q be the diameter of semicircle H. Circle O is internally tangent to H and tangent to P Q at C. Let A be a point on H and B a point on P Q such that AB is perpendicular to P Q and is also tangent to O. Prove that AC bisects ∠P AB. 