Constant Perimeter Triangle Theorem
If two tangents are drawn to a circle from a point A outside the circle and these tangents touch the circle at points B and C respectively, and a third tangent intersects AB in P and AC in R, and touches the circle at Q, then the perimeter of triangle APR = 2AB.
(Note: Quite surprisingly, and perhaps even counter-intuitively, it turns out that the perimeter of
triangle APR is only dependent on the length of AB and completely independent of not only the position and length of PR, but also the size of the circle).
Investigate
Drag point Q to move tangent PR, and the centre O to change the size of the circle. You can also change the length of AB by dragging either of X or Y (since XY was used to determine the length of AB).
Constant perimeter triangle formed by tangents to circle
Challenge
Can you explain why (prove that) the theorem is true?
Note: A special case of this little theorem was used as Problem no. 13 in the 2nd Round of the 2013 Senior South African Mathematics Olympiad (SAMO) - see De Villiers (2013) below.
------------------
A Slight Extension
Click on the 'Link to overlapping triangles' button to navigate to a new sketch.
A straight forward, direct application of the theorem is demonstrated in the new figure, which shows two triangles ABC and KLM overlapping, and circumscribed to the same circle. If either one, or both of these two triangles are rotated around the circle, then the perimeters of the coloured triangles APU, KQP, etc. remain constant (provided none of the points P, Q, R, etc. move onto the extensions (outside) of any of the sides of ABC and KLM).
Investigate
Drag point X or Y to rotate triangles ABC and KLM. You can also drag the point V to change the size of the circle. The lengths of the sides of triangles ABCand KLM can be changed by dragging any of the endpoints of segments AX, XB, KY or YL.
Reference
De Villiers, M. (2013). Reflecting on a 2nd Round 2013 SA Mathematics Olympiad Problem. Learning and Teaching Mathematics, No. 14, 2013, pp. 34-35.
Related Links
Logical Discovery: Circum Quad (Tangential quadrilateral) (Rethinking Proof Activity for Pitot's theorem)
Pitot's Theorem for a tangential/circumscribed quadrilateral (Same result as above without guided proof & some diagrams)
Logical Discovery: Varignon Parallelogram Perimeter (Rethinking Proof activity)
Circumscribed (Tangential) Hexagon Alternate Sides Theorem
The Tangential (or Circumscribed) Polygon Side Sum theorem
Theorem of Gusić & Mladinić
Converse of Tangent-Secant Theorem (Euclid Book III, Proposition 36)
SA Mathematics Olympiad 2016 Problem R2 Q20
A 1999 British Mathematics Olympiad Problem and its dual
Extangential Quadrilateral (Close nephew of circum quad)
Side Divider Theorem for a Circumscribed Quadrilateral
Conway's Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Perimeter inscribed parallel-hexagon
External Links
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
********************************
Back to "Dynamic Geometry Sketches"
Back to "Student Explorations"
Created by Michael de Villiers with JavaSketchpad, 20 April 2013; updated to WebSketchpad, 30 Oct 2025; 3 July 2026.