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 20070928 11:05:16 