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geo1 - Problems in Geometry(1 1 In a triangle ABC with...

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Problems in Geometry (1) 1. In a triangle ABC with orthocenter H and circumcircle O prove that < BAH = < OAC . 1 2. Let ABC be a triangle with orthocenter H in which AH, BH, CH meet BC, CA, AB in D, E, F respectively. Prove that HA · HD = HB · HE = HC · HF. 2 3. Let ABC be a triangle with orthocenter H in which AH, BH, CH meet BC, CA, AB in D, E, F respectively. (A ) If ABC is an acute angled triangle, prove that H is the incenter of DEF. 3 (B ) What happens if A is an obtuse angle ? 4. Let ABC be a triangle with incenter I and excenters I a , I b , I c . (A) Prove that I is the orthocenter of the triangle I a I b I c . (B) Prove that the circumcircle of a triangle passes through the midpoint of a line segment joining any two of the excenters. (C) Prove that the circumcircle of a triangle passes through the midpoint of a line segment joining the incenter with any one of the excenters. 1 Consider the right triangle BDA where D is the foot of the altitude through A and the isosceles triangle OCA.
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