Difference between revisions of "2021 USAJMO Problems/Problem 2"
Line 1: | Line 1: | ||
+ | ==Problem== | ||
Rectangles <math>BCC_1B_2,</math> <math>CAA_1C_2,</math> and <math>ABB_1A_2</math> are erected outside an acute triangle <math>ABC.</math> Suppose that<cmath>\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.</cmath>Prove that lines <math>B_1C_2,</math> <math>C_1A_2,</math> and <math>A_1B_2</math> are concurrent. | Rectangles <math>BCC_1B_2,</math> <math>CAA_1C_2,</math> and <math>ABB_1A_2</math> are erected outside an acute triangle <math>ABC.</math> Suppose that<cmath>\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.</cmath>Prove that lines <math>B_1C_2,</math> <math>C_1A_2,</math> and <math>A_1B_2</math> are concurrent. | ||
+ | |||
+ | ==Solution== | ||
+ | [[Image:Leonard my dude.png|frame|none|###px|]] | ||
+ | |||
+ | We first claim that the three circles <math>(BCC_1B_2),</math> <math>(CAA_1C_2),</math> and <math>(ABB_1A_2)</math> are share a common intersection. | ||
+ | |||
+ | Let the second intersection of <math>(BCC_1B_2)</math> and <math>(CAA_1C_2)</math> be <math>K</math>. Then | ||
+ | <cmath>\begin{align*} | ||
+ | \angle AKC &= 360^\circ - \angle BKA - \angle CKB \ | ||
+ | &= 360^\circ - (180^\circ - \angle AB_1B + 180^\circ - \angle BC_1C) \& | ||
+ | = 180^\circ - \angle CA_1A, | ||
+ | \end{align*}</cmath> | ||
+ | which implies that <math>AA_1C_2CK</math> is cyclic as desired. | ||
+ | |||
+ | Now we show that <math>K</math> is the intersection of <math>B_1C_2,</math> <math>C_1A_2,</math> and <math>A_1B_2.</math> Note that <math>\angle C_1XB = \angle BXA_2 = 90^\circ,</math> so <math>A_2, K, C_1</math> are collinear. Similarly, <math>B_1, K, C_2</math> and <math>A_1, K, B_2</math> are collinear, so the three lines concur and we are done. | ||
+ | |||
+ | ~Leonard_my_dude |
Revision as of 14:00, 15 April 2021
Problem
Rectangles
and
are erected outside an acute triangle
Suppose that
Prove that lines
and
are concurrent.
Solution
We first claim that the three circles
and
are share a common intersection.
Let the second intersection of and
be
. Then
which implies that
is cyclic as desired.
Now we show that is the intersection of
and
Note that
so
are collinear. Similarly,
and
are collinear, so the three lines concur and we are done.
~Leonard_my_dude