A parallelogram arising when erecting similar triangles on the sides of a quadrilateral—and corresponding historic notes

Q: What does this research mean for the field?

Erecting similar triangles on the sides of a quadrilateral results in the formation of a parallelogram, a geometric property that is accompanied by specific historical precedents. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

Hans Humenberger

doi:10.1080/00255572.2026.2714216

August 27, 2026The Mathematical GazetteOpen Access

A parallelogram arising when erecting similar triangles on the sides of a quadrilateral—and corresponding historic notes

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Key Points

Investigate the geometric conditions under which erecting similar triangles on the sides of a general quadrilateral generates a parallelogram, alongside its historical context.
Applied Euclidean geometric construction and synthetic proofs to configurations of similar triangles placed on quadrilateral edges.
Conducted a historical literature review tracing related quadrilateral theorems and polygon extension problems.
Showed that erecting similar triangles on the sides of an arbitrary quadrilateral consistently produces vertices or centers that form a parallelogram.
Outlined the evolution and historical lineage of side-erected polygon theorems across classical Euclidean geometry.

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Implication

Geometric analysis demonstrates parallelogram emergence from similar triangles erected on quadrilateral sides, highlighting historical developments in classical plane geometry.

Abstract

There are numerous interesting situations, problems and phenomena in which figures are erected on the sides of triangles or quadrilaterals. In many cases, the erected figures are similar to each ot...

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Hans Humenberger (2026) studied this question.

synapsesocial.com/papers/6a92995f8e5d7d1fc0c1138e https://doi.org/10.1080/00255572.2026.2714216

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A parallelogram arising when erecting similar triangles on the sides of a quadrilateral—and corresponding historic notes

Key Points

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Implication

Abstract

Discussion

Cite This Study

A parallelogram arising when erecting similar triangles on the sides of a quadrilateral—and corresponding historic notes

Key Points

Ask about this paper

Implication

Abstract

Discussion

Cite This Study

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