Proving with proportionality

BPT works inside one triangle. Many questions hide two triangles, and you use BPT in each one.

Step 1: find the triangle. Look at the parallel lines. One of them is part of a side of a bigger triangle, so follow that side round.

Step 2: link the two ratios. If two fractions are both equal to the same thing, they are equal to each other.

No triangle in sight? Make one. In a trapezium ABCD with AB ∥ DC, the parallel sides don't form a triangle. So join a diagonal, or draw a new parallel line through a point.

Example: the diagonals of a trapezium. The diagonals AC and BD meet at O. Draw EO through O parallel to DC (and to AB), meeting AD at E.

• In triangle ADC, EO ∥ DC, so AE/ED = AO/OC.
• In triangle DAB, EO ∥ AB, so AE/ED = BO/OD.
• Both equal AE/ED, so AO/OC = BO/OD.

So the diagonals of a trapezium cut each other in the same ratio. The converse is true as well: if they do, the quadrilateral is a trapezium.

Midpoints are a special case (ratio 1 : 1):

• a line through the midpoint of one side, parallel to another side, cuts the third side in half;
• the line joining the midpoints of two sides is parallel to the third side.

Checking with numbers: if AD = 6, DB = 9, AE = 8, EC = 12, then 6/9 = 2/3 and 8/12 = 2/3. The ratios are equal, so DE ∥ BC, and the corresponding angles are equal.