PQR is an acute angled triangle.

It is known that:

• S, T and U are points located respectively on the sides QR, RP and PQ.
• PS is perpendicular to QR.
• RU is the internal bisector of the angle PRQ.
• RU intersects PS and ST respectively at the points J and K.
• PT = TR.
• UJ=2, JK = 1 and KR = 3

Determine the perimeter of the triangle PQR.

Since K bisects UR and T bisects PR, TK is parallel to PQ. It follows that S bisects QR. So PS bisects angle QPR and the triangle is at least isosceles with apex at P and base QR.

Playing around with Geometer's Sketchpad, the triangle seems it must be uniquely equilateral. That is, only one position for Q relative to PR works, and since an equilateral triangle does work analytically, that is the unique solution.

Half of one side of the triangle comes out to 2*sqrt(3), so the full perimeter is 12*sqrt(3) ~= 20.78460969082653.

Posted by Charlie on 2007-09-28 11:05:16