By Yiu P.
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Example text
In this case, we call ABCD an orthocentric quadrangle. 4 Orthic triangle The orthic triangle of ABC has as vertices the traces of the orthocenter H on the sides. If ABC is an acute triangle, then the angles of the orthic triangle are 180◦ − 2α, 180◦ − 2β, and 180◦ − 2γ. A A Y Z Z H C X B B C X Y H If ABC is an obtuse triangle, with γ > 90◦ , then ABH is acute, with angles 90◦ − β, 90◦ − α, and 180◦ − γ. The triangles ABC and ABH have the same orthic triangle, whose angles are then 2β, 2α, and 2γ − 180◦ .
1 Let A, B, C, D be four points such that the lines AB and CD intersect (extended if necessary) at P . If AP · BP = CP · DP , then the points A, B, C, D are concyclic. 2 Let P be a point on the line containing the side AB of triangle ABC such that AP · BP = CP 2 . Then the line CP touches the circumcircle of triangle ABC at the point C. Exercise 1. Let ABC be a triangle satisfying γ = 90◦ + 12 β. If Z is the point on the side AB such that BZ = BC = a, then the circumcircle of triangle 45 P YIU: Euclidean Geometry 46 BCZ touches the side AC at C.
Then the internal common tangent of the two circles major passes through the midpoint of the arc AB. minor Proof. Let T be the point of contact, and CD the chord of (O) which is the internal common tangent of the circles K(P ) and K(Q). Regarding these two circles are tangent to the chord CD, and AB as an external common tangent, we conclude that C is the midpoint of the arc AB. 5 This leads to a simple construction of the two neighbors of (KP ), each tangent to (KP ), to the chord AB, and to the arc AB containing KP .
