A Lemma in Geometry Concerning Points Generated by the Incircle of a Triangle

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The incircle of triangle touches side
at
, and
is a diameter of the circle. If the line
meets
at
, then
.
Contributed by: Tomas Garza (December 2020)
Open content licensed under CC BY-NC-SA
Snapshots
Details
The situation is illustrated in step 1, where the line segment is a diameter of the incircle. The point
is determined by extending the segment
. The lemma states that
. Drag the Locator at
to modify the shape of the triangle.
The proof proceeds by making a translation of the incircle along line , so that point
is now placed on
, as shown in step 2. A dilation is now applied to the translated circle, keeping its top point fixed, until the dilated circle is tangent to the extended sides
and
(move the slider until the tangent points
and
appear in the figure). This is one of the excircles of the triangle.
This excircle and the incircle determine three vertices where the tangents property holds (i.e., two tangents to a circle from an external point are equal), namely (tangents to the excircle
and
, and tangents to the incircle
and
);
(tangents to the excircle
and
, and tangents to the incircle
and
); and
(tangents to the excircle
and
, and tangents to the incircle
and
).
From here on it is a matter of simple substitutions:
Starting from :
,
,
,
,
,
,
, Q.E.D.
Reference
[1] Y. Zhao. "Three Lemmas in Geometry." Massachusetts Institute of Technology Winter Camp 2010. yufeizhao.com/olympiad/three_geometry_lemmas.pdf.
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