Q739,738,736,7345,734,733,732,731,730

The figure shows a triangle ABC so that the angle C is double of angle A. If BD is the external angle bisector of angle B, prove that CD = AB + BC.

َ================================================= ============

Q738

In the diagram below, the circles O and O' intersect at A and B. CD is the common tangent. CA meets circle O' at F, DA meets circle O at G. GC and FD meet at H. BA meets GH at M. The circumcircles of triangles FGH and CHD meet at N. Prove that the points B, C, M, N are concyclic

------------------------------------------------------------------------------------------------------------------------------------------------------

Q737.

The figure shows a cyclic quadrilateral ABCD where AD is the diameter of the circumscribed circle. AB and DC extended meet at E. Tangents at B and C meet at F. Prove that EF is perpendicular to AD. This entry contributed by Ajit Athle

kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk kkkkkkkkkkk

Q736

The figure shows a triangle ABC with the altitude BH. HD is perpendicular to AB, and HE is perpendicular to BC. If angle ACB = 57 degrees, and angle CDH = 21 degrees, find the measure of the angle AEB.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Q735

The figure shows a triangle ABC with the altitudes AA1, BB1, and CC1. If B1D is perpendicular to AB, and B1E is perpendicular to BC, prove that DE is equal to the semiperimeter of the triangle A1B1C1.
Note: The triangle A1B1C1 is called the orthic triangle of the triangle ABC

++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++


Q734

The figure shows a triangle ABC with the median AM and a cevian BD so that BD = AD. If DE is parallel to AB, prove that the angles DBE and ACB are congruent. This entry contributed by Ajit Athle.

==kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk

Q733.

The figure shows a triangle ABC. L is a line through the orthocenter H that cuts the sides at A1, B1, and C1. Lines A1A2, B1B2, and C1C2 are the reflection of L in sides BC, AC, and AB, respectively. Prove that A1A2, B1B2, and C1C2 are concurrent at a point P on the circumcircle O

kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk kkk

Q732
The figure shows a triangle ABC with the altitudes AA1, BB1, and CC1. If A2, B2, and C2 are the orthocenters of triangles AB1C1, BA1C1, and CA1B1, respectively, prove that the triangles A1B1C1 and A2B2C2 are congruent.
Note: The triangle A1B1C1 is called the orthic triangle of the triangle ABC.

kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk kkkkkkkkkkkkkkkkkkk

Q731
The diagram shows a cyclic quadrilateral ABCD. If E is the orthocenter of triangle ABD and F the orthocenter of triangle ACD, prove that BCFE is a parallelogram

================================================== ===========================

Q730

The diagram shows a triangle ABC with the altitudes AA1, BB1, and CC1. A1A2, is perpendicular to AB, A1A3, is perpendicular to AC and AA4 is perpendicular to A2A3. Similarly BB4 is perpendicular to B2B3, and CC4 is perpendicular to C2C3. Prove that AA4, BB4 , and CC4 are concurrent.