Question

4. (a) Supply proof of the Menelaus Theorem concerning a transversal line L cutting the sides of ΔABC at points X,Y,Z respectively. (Hint) Drop perpendicular line segments from A, B, C to L and use similar triangles b)Centuries after Menelaus, Ceva discovered the Theorem that if P,Q, R are points on BC, CA and AB respectively so that AP, BQ, CR meet at a single point K, thern AR BP co RB PC QA Prove Ceva's theorem and its converse, by using Menelaus Theorem. (Hint: There are 6 triangles, each sharing a common side and a common angle with ΔABC, and each is being cut transversally by an appropriate line)

Answer 1

Let a triangle ABC with extended side BC.

Now Let a traversal line L cuts sides at X, Y and Z.

Now Let we drop perpendiculars from A, B and C to L.

From Similar triangles ---

$\frac{AZ}{BZ} = \frac{AP}{BQ}$ ______(1)
$\frac{BC}{CX} = \frac{BQ}{CR}$ ______(2)

$\frac{AY}{CY} = \frac{CR}{AP}$ ______(3)

Multiply all three equations ---

$\frac{AY}{CY} \times \frac{BC}{CX}\times \frac{AZ}{BZ} = \frac{CR}{AP} \times \frac{BQ}{CR} \times \frac{AP}{BQ}$

$\frac{AY}{CY} \times \frac{BC}{CX}\times \frac{AZ}{BZ} = 1$

Hence proved.