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In triangle ABC, point X is on side BC  such that AX=13, BX=14, CX=4,   and the circumcircles of triangles ABX and ACX have the same radius. Find the area of triangle ABC.

 May 30, 2020
 #1
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I placed the diagram on a coordinate axis, with P(0,0), C(4,0), and B(-14,0).

Since PA = 13, it is on a circle with center P and radius 13:  x2 + y2  =  13

-- I called the x-value of this point 'a', so the y-value became  sqrt(169 - a2).

 

I used the theorem that the center of a circle is on the perpendicular bisectors of the chords.

Since one chord is BP, its midpoint has x-value -7 and its y-value is on the line x = -7.

Since another chord is PC, its midpoint has x-value 2 and its y-value is on the line x = 2.

Another chord is AP: using its endpoints, I found its midpoint and its slope.

From these values, I could find the equation of its perpendicular bisector (not a lot of fun, for it had a lot of a-terms and square roots).

Then, I found the intersection of this perpendicular biscector with the line x = -7 and the intersection of this perpendicular bisector with the line x = 2 (there were still a lot of a-terms and square roots).

With these two points (the centers of the circumcircles), I could find the distance from one center to point P and the distance from the other center to point P (By this time, I was on a firt-name basis with the a-terms and square roots).

Since these are equal, by solving this equation, I could find the value of a, which was -14.

 

This is the x-value of point A, allowing me to find the y-value of point A, 15, which is the height of triangle ABC.

With the height 14 + 8 = 22 and the base 15, the area is 165.

 May 30, 2020
 #2
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165 is incorrect, a hint is "We're told that the circles have the same radius. In your diagram, what other parts of the circles must be equal? What does that tell you about triangle ABC?"

Guest May 30, 2020
 #3
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https://web2.0calc.com/questions/helpelp#r3

 May 31, 2020

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