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Difference between revisions of "1966 AHSME Problems/Problem 31"

(Created page with "== Problem == Triangle <math>ABC</math> is inscribed in a circle with center <math>O'</math>. A circle with center <math>O</math> is inscribed in triangle <math>ABC</math>. <math...")
 
(→‎See also)
 
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== Problem ==
 
== Problem ==
 +
<asy>
 +
draw(circle((0,0),10),black+linewidth(1));
 +
draw(circle((-1.25,2.5),4.5),black+linewidth(1));
 +
dot((0,0));
 +
dot((-1.25,2.5));
 +
draw((-sqrt(96),-2)--(-2,sqrt(96)),black+linewidth(.5));
 +
draw((-2,sqrt(96))--(sqrt(96),-2),black+linewidth(.5));
 +
draw((-sqrt(96),-2)--(sqrt(96)-2.5,7),black+linewidth(.5));
 +
draw((-sqrt(96),-2)--(sqrt(96),-2),black+linewidth(.5));
 +
MP("O'", (0,0), W);
 +
MP("O", (-2,2), W);
 +
MP("A", (-10,-2), W);
 +
MP("B", (10,-2), E);
 +
MP("C", (-2,sqrt(96)), N);
 +
MP("D", (sqrt(96)-2.5,7), NE);
 +
</asy>
 
Triangle <math>ABC</math> is inscribed in a circle with center <math>O'</math>. A circle with center <math>O</math> is inscribed in triangle <math>ABC</math>. <math>AO</math> is drawn, and extended to intersect the larger circle in <math>D</math>. Then we must have:
 
Triangle <math>ABC</math> is inscribed in a circle with center <math>O'</math>. A circle with center <math>O</math> is inscribed in triangle <math>ABC</math>. <math>AO</math> is drawn, and extended to intersect the larger circle in <math>D</math>. Then we must have:
  
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== Solution ==
 
== Solution ==
 +
We will prove that <math>\triangle DOB</math> and <math>\triangle COD</math> is isosceles, meaning that <math>CD=OD=BD</math> and hence <math>\fbox{D}</math>.
  
 +
Let <math>\angle A=2\alpha</math> and <math>\angle B=2\beta</math>. Since the incentre of a triangle is the intersection of its angle bisectors, <math>\angle OAB=\alpha</math> and <math>\angle ABO=\beta</math>. Hence <math>\angle DOB=\alpha +\beta</math>. Since quadrilateral <math>ABCD</math> is cyclic, <math>\angle CAD=\alpha=\angle CBD</math>. So <math>\angle OBD=\angle OBC+\angle CBD=\alpha + \beta = \angle DOB</math>. This means that <math>\triangle DOB</math> is isosceles, and hence <math>BD=OD</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>
  
 
== See also ==
 
== See also ==

Latest revision as of 17:52, 22 April 2024

Problem

[asy] draw(circle((0,0),10),black+linewidth(1)); draw(circle((-1.25,2.5),4.5),black+linewidth(1)); dot((0,0)); dot((-1.25,2.5)); draw((-sqrt(96),-2)--(-2,sqrt(96)),black+linewidth(.5)); draw((-2,sqrt(96))--(sqrt(96),-2),black+linewidth(.5)); draw((-sqrt(96),-2)--(sqrt(96)-2.5,7),black+linewidth(.5)); draw((-sqrt(96),-2)--(sqrt(96),-2),black+linewidth(.5)); MP("O'", (0,0), W); MP("O", (-2,2), W); MP("A", (-10,-2), W); MP("B", (10,-2), E); MP("C", (-2,sqrt(96)), N); MP("D", (sqrt(96)-2.5,7), NE); [/asy] Triangle $ABC$ is inscribed in a circle with center $O'$. A circle with center $O$ is inscribed in triangle $ABC$. $AO$ is drawn, and extended to intersect the larger circle in $D$. Then we must have:

$\text{(A) } CD=BD=O'D \quad \text{(B) } AO=CO=OD \quad \text{(C) } CD=CO=BD \\ \text{(D) } CD=OD=BD \quad \text{(E) } O'B=O'C=OD$

Solution

We will prove that $\triangle DOB$ and $\triangle COD$ is isosceles, meaning that $CD=OD=BD$ and hence $\fbox{D}$.

Let $\angle A=2\alpha$ and $\angle B=2\beta$. Since the incentre of a triangle is the intersection of its angle bisectors, $\angle OAB=\alpha$ and $\angle ABO=\beta$. Hence $\angle DOB=\alpha +\beta$. Since quadrilateral $ABCD$ is cyclic, $\angle CAD=\alpha=\angle CBD$. So $\angle OBD=\angle OBC+\angle CBD=\alpha + \beta = \angle DOB$. This means that $\triangle DOB$ is isosceles, and hence $BD=OD$.

Now let $\angle C=2\gamma$ which means $\angle ACO=COD=\gamma$. Since $ABCD$ is cyclic, $\angle DAB=\alpha=\angle DCB.$ Also, $\angle DAC\alpha$ so $\angle DOC=\alpha + \gamma$. Thus, $\angle OCD=\angle OCB+\angle BCD=\gamma + \alpha=\angle DOC$ which means $\triangle COD$ is isosceles, and hence $CD=OD=BD$.

Thus our answer is $\fbox{D.}$

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

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