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This paper derives properties of quadruples of nonnegative integers $(a,\\, b,\\, c,\\, d)$, called triangle quadruples, satisfying this equation. It is easy to verify that the operation generating $(a,\\, b,\\, c,\\, a+b+c-d)$ from $(a,\\, b,\\, c,\\, d)$ preserves this feature and that it and analogous ones for the other elements "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1303.0203","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-03-01T15:53:27Z","cross_cats_sorted":[],"title_canon_sha256":"f77722ac663727f628d41d2d202a56c07f051fd3eea298bf8eaf9bb34f2644c7","abstract_canon_sha256":"e0d606b742bc2e3f41863cb56114503f8e64e582843325fe0cb650872ee9a18c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:32:06.569763Z","signature_b64":"ZHU3F8ks83eSTOT+iHXh3b3YU2drCSnMzApyDcrOSXKn0hidfKEHW6CYvGK1vCKFZlhPbhNOIlqHo74R8AawAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"67e287fc2110a61ae65e88c186f40815fed5a1df47be3682c905b1b71394c8cf","last_reissued_at":"2026-05-18T03:32:06.568976Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:32:06.568976Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Apollonian Equilateral Triangles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Christina Chen, Nan Li","submitted_at":"2013-03-01T15:53:27Z","abstract_excerpt":"Given an equilateral triangle with $a$ the square of its side length and a point in its plane with $b$, $c$, $d$ the squares of the distances from the point to the vertices of the triangle, it can be computed that $a$, $b$, $c$, $d$ satisfy $3(a^2+b^2+c^2+d^2)=(a+b+c+d)^2$. 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