# Angles on Equal Arcs are Equal

Jump to navigation Jump to search

## Theorem

In equal circles, angles standing on equal arcs are equal to one another, whether at the center or at the circumference of those circles.

In the words of Euclid:

In equal circles angles standing on equal circumferences are equal to one another, whether they stand at the centres or at the circumferences.

## Proof

Let $ABC$ and $DEF$ be equal circles.

Let $\angle BGC$ and $\angle EHF$ stand on the equal arcs $BC$ and $EF$.

Suppose $\angle BGC \ne \angle EHF$.

Then one of them is bigger.

Suppose $\angle BGC > \angle EHF$.

From Equal Angles in Equal Circles, these angles stand on equal arcs.

So arcs $BK = EF$.

But $EF = BC$ and so $BK = BC$, which is impossible.

So $\angle BGC$ and $\angle EHF$ are not unequal, therefore $\angle BGC = \angle EHF$.

From the Inscribed Angle Theorem, $2 \angle BAC = \angle BGC$ and $2 \angle EDF = \angle EHF$.

Therefore $2 \angle BAC = \angle EHF$.

$\blacksquare$

## Historical Note

This proof is Proposition $27$ of Book $\text{III}$ of Euclid's The Elements.
It is the converse of Proposition $26$: Equal Angles in Equal Circles.
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://proofwiki.org/wiki/Angles_on_Equal_Arcs_are_Equal", "fetch_time": 1679439778000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2023-14/segments/1679296943747.51/warc/CC-MAIN-20230321225117-20230322015117-00673.warc.gz", "warc_record_offset": 534952526, "warc_record_length": 11674, "token_count": 321, "char_count": 1168, "score": 4.34375, "int_score": 4, "crawl": "CC-MAIN-2023-14", "snapshot_type": "latest", "language": "en", "language_score": 0.7501713037490845, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00027-of-00064.parquet"}