Difference between revisions of "1966 AHSME Problems/Problem 31"
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Ruohongzhou (talk | contribs) (→Solution) |
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Now let <math>\angle C=2\gamma</math> which means <math>angle ACO=COD=\gamma</math>. Since <math>ABCD</math> is cyclic, <math>\angle DAB=\alpha=\angle DCB.</math> Also, <math>\angle DAC\alpha</math> so <math>\angle DOC=\alpha + \gamma</math>. Thus, <math>\angle OCD=\angle OCB+\angle BCD=\gamma + \alpha=\angle DOC</math> which means <math>\triangle COD</math> is isosceles, and hence <math>CD=OD=BD</math>. | Now let <math>\angle C=2\gamma</math> which means <math>angle ACO=COD=\gamma</math>. Since <math>ABCD</math> is cyclic, <math>\angle DAB=\alpha=\angle DCB.</math> Also, <math>\angle DAC\alpha</math> so <math>\angle DOC=\alpha + \gamma</math>. Thus, <math>\angle OCD=\angle OCB+\angle BCD=\gamma + \alpha=\angle DOC</math> which means <math>\triangle COD</math> is isosceles, and hence <math>CD=OD=BD</math>. | ||
− | Thus our answer is <math>\fbox{D}</math> | + | Thus our answer is <math>\fbox{D}</math>!!! |
== See also == | == See also == |
Revision as of 20:45, 20 June 2020
Problem
Triangle
is inscribed in a circle with center
. A circle with center
is inscribed in triangle
.
is drawn, and extended to intersect the larger circle in
. Then we must have:
Solution
We will prove that and
is isosceles, meaning that
and hence
.
Let and
. Since the incentre of a triangle is the intersection of its angle bisectors,
and
. Hence
. Since quadrilateral
is cyclic,
. So
. This means that
is isosceles, and hence
.
Now let which means
. Since
is cyclic,
Also,
so
. Thus,
which means
is isosceles, and hence
.
Thus our answer is !!!
See also
1966 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 30 |
Followed by Problem 32 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.