225. The Midpoint Bridge · parallel, proportional, and perfectly half

Try to break it

Drag A, B, or C. D and E sit at the midpoints of AB and AC. DE always stays parallel to BC and measures exactly half its length. Pull the triangle into any shape — tall, flat, or skewed — DE = BC/2 every time.

How you build it

Make a triangle with its midpoint segment.

• Place point A as the apex of the triangle.
• Place point B as the bottom-left vertex.
• Place point C as the bottom-right vertex.
• Draw segment AB.
• Draw segment BC.
• Draw segment CA to complete triangle ABC.
• Midpoint tool: click A then B — D is dropped exactly at the midpoint of AB.
• Midpoint tool: click A then C — E is dropped exactly at the midpoint of AC.
• Draw segment DE connecting the midpoints.

The proof, step by step

Prove that the segment joining the midpoints of two sides is parallel to the third side and half its length.

• Extend DE to F such that EF = DE. Join CF.
• In ΔADE and ΔCFE: AE = EC (E is midpoint), ∠AED = ∠CEF (vertically opposite), DE = EF (by construction). So ΔADE ≅ ΔCFE (SAS).
• Thus AD = CF and ∠DAE = ∠FCE. Since alternate angles are equal, AB // CF.
• Since AD = BD (D is midpoint), we have BD = CF. With BD // CF, BCFD is a parallelogram.
• In parallelogram BCFD, DF // BC and DF = BC. Since DE = ½ DF, we conclude DE = ½ BC and DE // BC.

Worked example

In ΔPQR, S and T are midpoints of PQ and PR respectively. If QR = 18 cm, find the length of ST.

By the Mid-point Theorem, the segment joining the midpoints of two sides is parallel to the third side and half its length. ST = ½ × QR = ½ × 18 = 9 cm.

• 4.5 cm
• 9 cm — correct
• 18 cm
• 36 cm