2000 JBMO Problems/Problem 3
Problem 3
A half-circle of diameter is placed on the side
of a triangle
and it is tangent to the sides
and
in the points
and
respectively. Prove that the intersection point
between the lines
and
lies on the altitude from
of the triangle
.
Solution
Let be the midpoint of diameter
.
Let
meet
at
.
We begin by showing that is the circumcenter of
:
Let us define and
By applying Tangent Chord Angle theorem, we get:
and
Now, , and since
is a cyclic quadrilateral,
we have
Now , so
Similarly, we have , so
From
Thus, we have
Also, (Since
and
are tangents to the same circle)
From the above 2 results, it readily follows that is the circumcenter of
.
Thus, we have , and so
So in
So is perpendicular to
, hence
lies on the altitude from
of the triangle
.