ratios AM MB AN NC and AR RB AS SC Check if they are equal? The conclusion will lead us to one of the most important theorem in Geometry, which we will discuss below. Theorem : Basic Proportionality Theorem (BPT) or Thales theorem Statement A straight line drawn parallel to a side of triangle intersecting the other two sides, divides the sides in the same ratio. Proof Given: In D ABC , D is a point on AB and E is a point on AC .
To prove: AD DB AE EC Construction: Draw a line DE BC No. Statement Reason . ∠ = ∠ = ∠ ABC ADE Corresponding angles are equal because DE BC . ∠ = ∠ = ∠ ACB AED Corresponding angles are equal because DE BC .
∠ = ∠ = ∠ DAE BAC Both triangles have a common angle . D D ABC ADE AB AD AC AE AD DB AD AE EC AE + DB AD = + EC AE DB AD EC AE AD DB AE EC By AAA similarity Corresponding sides are proportional Split AB and AC using the points D and E . On simplification Cancelling on both sides Taking reciprocals Hence proved M P R N Q S Fig. .
D E Fig. . Geometry Corollary If in D ABC , a straight line DE parallel to BC , intersects AB at D and AC at E , then (i) AB AD AC AE (ii) AB DB AC EC Proof In D ABC , DE BC Therefore, AD DB AE EC (by Basic Proportionality Theorem) (i) Taking reciprocals, we get DB AD EC AE Add to both in the sides DB AD EC AE DB AD AD EC AE AE so, AB AD AC AE (ii) Add to both the sides AD DB AE EC Therefore, AB DB AC EC Is the converse of Basic Proportionality Theorem also true? To examine let us do the following illustration.
Illustration