Proof
Let be bisected at (I.10) and produced to . At erect
at right angles to (I.11), with . Join ,
, . Through draw parallel to , of length such
that meets the line through parallel to at (I.31).
As in II.9, is a right angle (right-isoceles triangles
and ). Triangle is also right-angled, with the
right angle at in the configuration where lies on the
extension of . Apply I.47 twice (to and to the
triangle formed by extending the constructions to ):
\[
AD^2 + DB^2 \;=\; 2\cdot AC^2 + 2\cdot CD^2,
\]
where now is the half plus the added segment. The
derivation parallels II.9 exactly with extension in place of internal
cut, and the symmetry is what Heath emphasises.
Knowledge graph · drag to pan, scroll to zoom, click a node to navigate
Full neighborhood
Depends on (10)
- II.9Proposition II.9If a straight line be cut into equal and unequal segments, the squares on the unequal segments of the whole are double…
- I.5Proposition I.5In isosceles triangles the angles at the base are equal to one another; and if the equal straight lines be produced…
- I.10Proposition I.10To bisect a given finite straight line.
- I.11Proposition I.11To draw a straight line at right angles to a given straight line from a given point on it.
- I.29Proposition I.29A straight line falling on parallel straight lines makes the alternate angles equal to one another, the exterior angle…
- I.31Proposition I.31Through a given point to draw a straight line parallel to a given straight line.
- I.32Proposition I.32In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles,…
- I.47Proposition I.47In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides…
- 2Common notion 2If equals be added to equals, the wholes are equal.
- 3Common notion 3If equals be subtracted from equals, the remainders are equal.
Discussion
No replications, contradictions, or comments registered yet for this claim.
Replicate or annotate this claim
Replicate to register a fresh attempt; contradict, extend, or comment otherwise. Authors can post a claim-retraction with the reason taxonomy from RRP-0020.
Sign in with ORCID to annotate this claim.