Passel of Techniques -- 2
Let
be a triangle in which
. Suppose the
orthocenter
of the triangle lies on the incircle, find the ratio
.
If this ratio can be expressed as , where and are coprime positive integers, submit your answer as .
The answer is 7.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
Let G be the midpoint of DC. Join G to I, the in-centre and extend it to meet AB in J. We note that GI//CH and thus meets AB at a right angle. Moreover, J is thus the point of tangency of the in-circle with side AB. We further claim that triangles BGJ & ABD are similar and therefore, AB/BD = BG/BJ =BG/BD. Thus, AB=BG=BC/2+BC/4=(3/4)BC which yields, AB/BC = 3/4 or p+q =7..