Difference between revisions of "2023 CMO Problems/Problem 5"
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In an acute triangle <math>\triangle A B C, K</math> is a point on the extension of <math>B C</math>. Through <math>K</math>, draw lines parallel to <math>A B</math> and <math>A C</math>, denoted as <math>K P</math> and <math>K Q</math> respectively, such that <math>B K=B P</math> and <math>C K=C Q</math>. Let the circumcircle of <math>\triangle K P Q</math> intersect <math>A K</math> at point <math>T</math>. Prove: | In an acute triangle <math>\triangle A B C, K</math> is a point on the extension of <math>B C</math>. Through <math>K</math>, draw lines parallel to <math>A B</math> and <math>A C</math>, denoted as <math>K P</math> and <math>K Q</math> respectively, such that <math>B K=B P</math> and <math>C K=C Q</math>. Let the circumcircle of <math>\triangle K P Q</math> intersect <math>A K</math> at point <math>T</math>. Prove: | ||
Latest revision as of 04:36, 25 May 2024
Problem
In an acute triangle is a point on the extension of
. Through
, draw lines parallel to
and
, denoted as
and
respectively, such that
and
. Let the circumcircle of
intersect
at point
. Prove:
(1) ;
(2) .
Solution 1
Proof for (1):
Let the side lengths of be
. We have
.
Let
.
Assume :
Assume :
Assume
Proof: (2)
~xiaohuangya|szm
See Also
2023 CMO(CHINA) (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All CMO(CHINA) Problems and Solutions |