Math 2.0

Geometry Proof

Proof of Apollonius's theorem from the parallelogram law

Builds on MidpointEuclidean distanceLength of a vectorParallelogram law

Proof

We write the two sides from AA as a sum and a difference of two vectors.

  1. By the definition of the midpoint, D=B+12(C−B)D = B + \frac{1}{2}(C - B).
  2. Let m=D−Am = D - A and w=B−Dw = B - D. By step 1, w=−12(C−B)w = -\frac{1}{2}(C - B) and C−D=12(C−B)=−wC - D = \frac{1}{2}(C - B) = -w.
  3. Then B−A=(D−A)+(B−D)=m+wB - A = (D - A) + (B - D) = m + w and C−A=(D−A)+(C−D)=m−wC - A = (D - A) + (C - D) = m - w.
  4. With the Euclidean distance and the length, ∣AB∣=∥m+w∥\abs{AB} = \norm{m + w}, ∣AC∣=∥m−w∥\abs{AC} = \norm{m - w}, ∣AD∣=∥m∥\abs{AD} = \norm{m} and ∣BD∣=∥w∥\abs{BD} = \norm{w}.
  5. ℝn\R^n with the dot product is an inner product space with the norm ∥⋅∥\norm{\cdot}. By the parallelogram law, ∥m+w∥2+∥m−w∥2=2∥m∥2+2∥w∥2\norm{m + w}^2 + \norm{m - w}^2 = 2 \norm{m}^2 + 2 \norm{w}^2.
  6. By step 4, this is ∣AB∣2+∣AC∣2=2(∣AD∣2+∣BD∣2)\abs{AB}^2 + \abs{AC}^2 = 2 \left( \abs{AD}^2 + \abs{BD}^2 \right).

Idea

Seen from AA, the vertices BB and CC sit at m+wm + w and m−wm - w. Here mm goes to the midpoint and ww is half the side BCBC. The parallelogram law for mm and ww is then exactly the theorem.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first. The path takes the proof Proof of the parallelogram law.

Show the path as a graph
Euclidean distance, concept, no evidence levelEuclidean distanceconceptInner product, concept, no evidence levelInner productconceptInner product space, concept, no evidence levelInner product spaceconceptLength of a vector, concept, no evidence levelLength of a vectorconceptMidpoint, concept, no evidence levelMidpointconceptNorm induced by an inner product, concept, no evidence levelNorm induced by an in…conceptVector space, definition, no evidence levelVector spacedefinitionProof of the parallelogram law, proof, no evidence levelProof of the parallel…proofInner product, definition, no evidence levelInner productdefinitionNorm, definition, no evidence levelNormdefinitionInner product space, definition, no evidence levelInner product spacedefinitionNorm induced by an inner product, definition, no evidence levelNorm induced by an in…definitionParallelogram law, statement, no evidence levelParallelogram lawstatementProof of Apollonius's theorem from the parallelogram law, proof, no evidence levelProof of Apollonius's…proof
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean distance concept
  2. Inner product concept
  3. Inner product space concept
  4. Length of a vector concept
  5. Midpoint concept
  6. Norm induced by an inner product concept
  7. Vector space definition
  8. Proof of the parallelogram law proof
  9. Inner product definition
  10. Norm definition
  11. Inner product space definition
  12. Norm induced by an inner product definition
  13. Parallelogram law statement
  14. Proof of Apollonius's theorem from the parallelogram law proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Apollonius%27s_theorem, Statement and relation to other theorem
About this page
TagB0DPYBE7XM
URIm2://B0DPYBE7XM
Kindproof
WritingW0 stub
MSC51M04
ProvesApollonius's theorem
UsesMidpoint, Euclidean distance, Length of a vector, Parallelogram law
Changed4 November 2026

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