Segar Rogers
@segarrogers.bsky.social
Teacher. Maths. Secondary. Edinburgh.
Old enough to remember chalk.
Poetry on a Sunday.
Old enough to remember chalk.
Poetry on a Sunday.
Trying to write a question that requires the use of the Angle Bisector Theorem (Eu. El. VI – 3) … but failing … somehow a non VI – 3 solution always presents itself. Still, ended up with a nice enough question.
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November 3, 2025 at 1:08 PM
Trying to write a question that requires the use of the Angle Bisector Theorem (Eu. El. VI – 3) … but failing … somehow a non VI – 3 solution always presents itself. Still, ended up with a nice enough question.
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Does anyone teach this? Wondering if it's generally known or generally not known.
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November 1, 2025 at 5:38 PM
Does anyone teach this? Wondering if it's generally known or generally not known.
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I thought your very beautiful solution deserved to be drawn :-)
October 27, 2025 at 8:26 PM
I thought your very beautiful solution deserved to be drawn :-)
I'm stuck too. Going round in circles ... struggling to prove that the red line goes through the red dot (which would lead on to a = b = 22½).
October 26, 2025 at 12:26 PM
I'm stuck too. Going round in circles ... struggling to prove that the red line goes through the red dot (which would lead on to a = b = 22½).
An hour in the wind this afternoon collecting an oak forest :-)
October 25, 2025 at 5:42 PM
An hour in the wind this afternoon collecting an oak forest :-)
Nice. With fractions I think of the denominator as a noun.
October 20, 2025 at 7:33 PM
Nice. With fractions I think of the denominator as a noun.
Been thinking more about the merits of the Exterior Angle Theorem and the fluency of pupils' geometric thinking. Here's a proof of one of @karencampe.bsky.social 's angle–arc theorems. Step 4 is beautifully immediate _if_ one uses Exterior-Angle-Theorem thinking.
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October 20, 2025 at 5:24 PM
Been thinking more about the merits of the Exterior Angle Theorem and the fluency of pupils' geometric thinking. Here's a proof of one of @karencampe.bsky.social 's angle–arc theorems. Step 4 is beautifully immediate _if_ one uses Exterior-Angle-Theorem thinking.
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Zeta CfE Third Level.
October 13, 2025 at 5:53 PM
Zeta CfE Third Level.
Left: Requires triangle-interior-angle-sum and straight-angle-sum. 3 steps. Slower?
Right: Exterior Angle Theorem (Euclid 1–32). Requires corresponding and alternate angles. 1 step. Faster?
Right feels cleaner to me ... I wonder what pupils would think?
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Right: Exterior Angle Theorem (Euclid 1–32). Requires corresponding and alternate angles. 1 step. Faster?
Right feels cleaner to me ... I wonder what pupils would think?
#UKMathsChat #iTeachMaths
October 13, 2025 at 1:47 PM
Left: Requires triangle-interior-angle-sum and straight-angle-sum. 3 steps. Slower?
Right: Exterior Angle Theorem (Euclid 1–32). Requires corresponding and alternate angles. 1 step. Faster?
Right feels cleaner to me ... I wonder what pupils would think?
#UKMathsChat #iTeachMaths
Right: Exterior Angle Theorem (Euclid 1–32). Requires corresponding and alternate angles. 1 step. Faster?
Right feels cleaner to me ... I wonder what pupils would think?
#UKMathsChat #iTeachMaths
This one is a little harder to see because the vertex of the 55° angle coincides with the end of the 110° arc. But the 110° arc is still being subtended from a point on the circumference, so Eu III-20 still applies.
6/6
6/6
October 9, 2025 at 11:37 AM
This one is a little harder to see because the vertex of the 55° angle coincides with the end of the 110° arc. But the 110° arc is still being subtended from a point on the circumference, so Eu III-20 still applies.
6/6
6/6
... and the same again, this time starting at the 40°
1 → 2: Eu. III, 20
2 → 3: Supplementary arc.
3 → 4: Eu. III, 20
5/6
1 → 2: Eu. III, 20
2 → 3: Supplementary arc.
3 → 4: Eu. III, 20
5/6
October 9, 2025 at 11:37 AM
... and the same again, this time starting at the 40°
1 → 2: Eu. III, 20
2 → 3: Supplementary arc.
3 → 4: Eu. III, 20
5/6
1 → 2: Eu. III, 20
2 → 3: Supplementary arc.
3 → 4: Eu. III, 20
5/6
Start at the 20° and follow the sequence of steps.
1 → 2: Eu. III-20
2 → 3: Supplementary arc.
3 → 4: Eu. III-20
4/6
1 → 2: Eu. III-20
2 → 3: Supplementary arc.
3 → 4: Eu. III-20
4/6
October 9, 2025 at 11:37 AM
Start at the 20° and follow the sequence of steps.
1 → 2: Eu. III-20
2 → 3: Supplementary arc.
3 → 4: Eu. III-20
4/6
1 → 2: Eu. III-20
2 → 3: Supplementary arc.
3 → 4: Eu. III-20
4/6
Euclid's Elements Book 3 Proposition 20 (called by some the 'Inscribed Angle Theorem') will look like this: the arc is double the angle (on the circumference) that subtends it.
3/6
3/6
October 9, 2025 at 11:37 AM
Euclid's Elements Book 3 Proposition 20 (called by some the 'Inscribed Angle Theorem') will look like this: the arc is double the angle (on the circumference) that subtends it.
3/6
3/6
An arc of 30° will have a supplementary arc of 150°.
2/6
2/6
October 9, 2025 at 11:37 AM
An arc of 30° will have a supplementary arc of 150°.
2/6
2/6
A very short primer on mixing degrees of arc with degrees of angle for circle theorems.
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Historically degrees measured arcs and arcs measured angles. This might feel uncomfortable but remember how we measure an angle today … with a protractor … measuring around an arc.
1/6
#UKMathsChat #iTeachMath
Historically degrees measured arcs and arcs measured angles. This might feel uncomfortable but remember how we measure an angle today … with a protractor … measuring around an arc.
1/6
October 9, 2025 at 11:37 AM
A very short primer on mixing degrees of arc with degrees of angle for circle theorems.
#UKMathsChat #iTeachMath
Historically degrees measured arcs and arcs measured angles. This might feel uncomfortable but remember how we measure an angle today … with a protractor … measuring around an arc.
1/6
#UKMathsChat #iTeachMath
Historically degrees measured arcs and arcs measured angles. This might feel uncomfortable but remember how we measure an angle today … with a protractor … measuring around an arc.
1/6
Yes, exactly. But try to avoid converting it back to what you know and work on the circumference rather than at the centre; all you really need to know is Euclid III, 21.
October 8, 2025 at 7:41 PM
Yes, exactly. But try to avoid converting it back to what you know and work on the circumference rather than at the centre; all you really need to know is Euclid III, 21.
Something I didn't know.
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August 29, 2025 at 8:05 AM
Something I didn't know.
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June 29, 2025 at 8:55 AM
Euclid. Elements. 1.47. I see what you mean!
June 14, 2025 at 5:15 PM
Euclid. Elements. 1.47. I see what you mean!
... just realised ... the orientation does have the merit of leading nicely to the basis of the Sine Rule ... i.e. a similar orientation.
June 14, 2025 at 10:42 AM
... just realised ... the orientation does have the merit of leading nicely to the basis of the Sine Rule ... i.e. a similar orientation.
An aide-memoire for right-angled trig?! … maybe not … but it's interesting how orientation affects our ability to remember relationships. In this orientation I find the symmetry of the relationships much clearer. If only all trig questions had horizontal hypotenuses ;-)
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June 14, 2025 at 10:03 AM
An aide-memoire for right-angled trig?! … maybe not … but it's interesting how orientation affects our ability to remember relationships. In this orientation I find the symmetry of the relationships much clearer. If only all trig questions had horizontal hypotenuses ;-)
#UKMathsChat #iTeachMaths
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Went round in circles with this many times ... could never quite prove to myself that the purple line has the same gradient as the black dashed line ... many false assumptions and many erroneous deductions! However, I think (*think*) this does it.
June 8, 2025 at 12:13 PM
Went round in circles with this many times ... could never quite prove to myself that the purple line has the same gradient as the black dashed line ... many false assumptions and many erroneous deductions! However, I think (*think*) this does it.