Difference between revisions of "2011 AMC 12B Problems/Problem 20"

Revision as of 16:59, 14 July 2021

Problem

Triangle $ABC$ has $AB = 13, BC = 14$ , and $AC = 15$ . The points $D, E$ , and $F$ are the midpoints of $\overline{AB}, \overline{BC}$ , and $\overline{AC}$ respectively. Let $X \neq F$ be the intersection of the circumcircles of $\triangle BDE$ and $\triangle CEF$ . What is $XA + XB + XC$ ?

$\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 14\sqrt{3} \qquad \textbf{(C)}\ \frac{195}{8} \qquad \textbf{(D)}\ \frac{129\sqrt{7}}{14} \qquad \textbf{(E)}\ \frac{69\sqrt{2}}{4}$

Solutions

Solution 1 (Coordinates)

Let us also consider the circumcircle of $\triangle ADF$ .

Note that if we draw the perpendicular bisector of each side, we will have the circumcenter of $\triangle ABC$ which is $P$ , Also, since $m\angle ADP = m\angle AFP = 90^\circ$ . $ADPF$ is cyclic, similarly, $BDPE$ and $CEPF$ are also cyclic. With this, we know that the circumcircles of $\triangle ADF$ , $\triangle BDE$ and $\triangle CEF$ all intersect at $P$ , so $P$ is $X$ .

The question now becomes calculate the sum of distance from each vertices to the circumcenter.

We can calculate the distances with coordinate geometry. (Note that $XA = XB = XC$ because $X$ is the circumcenter.)

Let $A = (5,12)$ , $B = (0,0)$ , $C = (14, 0)$ , $X= (x_0, y_0)$

Then $X$ is on the line $x = 7$ and also the line with slope $-\frac{5}{12}$ that passes through $(2.5, 6)$ .

$y_0 = 6-\frac{45}{24} = \frac{33}{8}$

So $X = (7, \frac{33}{8})$

and $XA +XB+XC = 3XB = 3\sqrt{7^2 + \left(\frac{33}{8}\right)^2} = 3\times\frac{65}{8}=\frac{195}{8}$

Solution 2 (Algebra)

Consider an additional circumcircle on $\triangle ADF$ . After drawing the diagram, it is noticed that each triangle has side values: $7$ , $\frac{15}{2}$ , $\frac{13}{2}$ . Thus they are congruent, and their respective circumcircles are. By inspection, we see that $XA$ , $XB$ , and $XC$ are the circumdiameters, and so they are congruent. Therefore, the solution can be found by calculating one of these circumdiameters and multiplying it by a factor of $3$ . We can find the circumradius quite easily with the formula $\sqrt{(s)(s-a)(s-b)(s-c)} = \frac{abc}{4R}$ , such that $s=\frac{a+b+c}{2}$ and $R$ is the circumradius. Since $s = \frac{21}{2}$ :

$\sqrt{(\frac{21}{2})(4)(3)(\frac{7}{2})} = \frac{\frac{15}{2}\cdot\frac{13}{2}\cdot 7}{4R}$

After a few algebraic manipulations:

$\Rightarrow R=\frac{65}{16} \Rightarrow XA = XB = XC = \frac{65}{8} \Rightarrow XA + XB + XC = \boxed{\frac{195}{8}}$ .

Solution 3 (Homothety)

Let $O$ be the circumcenter of $\triangle ABC,$ and $h_A$ denote the length of the altitude from $A.$ Note that a homothety centered at $B$ with ratio $\frac{1}{2}$ takes the circumcircle of $\triangle BAC$ to the circumcircle of $\triangle BDE$ . It also takes the point diametrically opposite $B$ on the circumcircle of $\triangle BAC$ to $O.$ Therefore, $O$ lies on the circumcircle of $\triangle BDE.$ Similarly, it lies on the circumcircle of $\triangle CEF.$ By Pythagorean triples, $h_A=12.$ Finally, our answer is $3R=3\cdot \frac{abc}{4\{ABC\}}=3\cdot \frac{abc}{2ah_A}=3\cdot \frac{bc}{2h_A}=\boxed{\frac{195}{8}.}$

Solution 4 (basically Solution 1 but without coordinates)

Since Solution 1 has already proven that the circumcenter of $\triangle ABC$ coincides with $X$ , we'll go from there. Note that the radius of the circumcenter of any given triangle is $\frac{a}{2\sin{A}}$ , and since $b=15$ and $\sin{B}=\frac{12}{13}$ , it can be easily seen that $XA = XB = XC = \frac{65}{8}$ and therefore our answer is $3\cdot \frac{65}{8}=\boxed{\frac{195}{8}}.$

@@ Line 32: / Line 32: @@
 After a few algebraic manipulations:
-<math>\Rightarrow R=\frac{65}{8} \Rightarrow 3R = \boxed{\frac{195}{8}}</math>.
+<math>\Rightarrow R=\frac{65}{16} \Rightarrow XA = XB = XC = \frac{65}{8} \Rightarrow XA + XB + XC = \boxed{\frac{195}{8}}</math>.
 ===Solution 3 (Homothety)===

2011 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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