Difference between revisions of "2002 Pan African MO Problems/Problem 5"
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Latest revision as of 12:47, 27 May 2024
Problem
Let be an acute angled triangle. The circle with diameter
intersects the sides
and
at points
and
respectively. The tangents drawn to the circle through
and
intersect at
.
Show that
lies on the altitude through the vertex
.
Solution
Draw lines
and
, where
and
are on
and
, respectively. Because
and
are tangents as well as
and
,
and
. Additionally, because
and
are tangents,
.
Let and
. By the Base Angle Theorem,
and
. Additionally, from the property of tangent lines,
,
,
, and
. Thus, by the Angle Addition Postulate,
and
. Thus,
and
, so
. Since the sum of the angles in a quadrilateral is 360 degrees,
. Additionally, by the Vertical Angle Theorem,
and
. Thus,
.
Now we need to prove that
is the center of a circle that passes through
. Extend line
, and draw point
not on
such that
is on the circle with
. By the Triangle Angle Sum Theorem and Base Angle Theorem,
. Additionally, note that
, and since
,
. Thus, by the Base Angle Converse,
. Furthermore,
. Therefore,
is the diameter of the circle, making
the radius of the circle. Since
is a point on the circle,
.
Thus, by the Base Angle Theorem, , so
. Since
, by the Alternating Interior Angle Converse,
. Therefore, since
,
, and
must be on the altitude of
that is through vertex
.
Solution 2 (by duck_master)
Let be the intersection of
and
. Note that
, and similarly
. Thusly,
is a cyclic quadrilateral, and
is the diameter of its circumcircle.
Next, let be the intersection of
and
; we claim that
. Note that
, so
is cyclic. Then
, so
.
Furthermore, we claim that is the midpoint of
. To show this, we use the method of phantom points: we let
be the midpoint of
. Then
, and
. Since the two values match, we have
. Similarly, we show that
. This necessarily implies
.
Finally, we show that lies on the height from
to
. Since
, we know that
is the height from
to
. But
, so
lies on
and we are done.
See Also
2002 Pan African MO (Problems) | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
All Pan African MO Problems and Solutions |