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Problems on tangents: circles inside triangles and quadrilaterals sheet
3-mark answer frame

The sides AB, BC, CD and DA of a quadrilateral ABCD touch a circle at P, Q, R and S. Prove that AB + CD = AD + BC.

How to lay out the answer. Follow the frame, then write it in your own words.

  1. 1 · Write what is given and what is to be found Given: the sides of ABCD touch a circle at P (on AB), Q (on BC), R (on CD) and S (on DA). To prove: AB + CD = AD + BC.
  2. 2 · Draw the figure Circle inside ABCD with the four points of contact marked; tick the equal tangents.
  3. 3 · Write the formula or theorem Tangents drawn from an external point to a circle are equal.
  4. 4 · Write the equal pairs AP = AS, BP = BQ, CR = CQ, DR = DS.
  5. 5 · Add them AP + BP + CR + DR = AS + BQ + CQ + DS.
  6. 6 · Group the terms (AP + BP) + (CR + DR) = (AS + DS) + (BQ + CQ), so AB + CD = AD + BC.
  7. 7 · Check and box the answer Write 'Hence proved'.

Scoring tip: Write the four equal pairs in one line with the reason 'tangents from an external point'. The grouping step then follows in one line.