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Geometry Problem 1

Prove that in a triangle, the Nagel point and incenter concur if and only if the triangle is equilateral.

\( \textbf{Note:} \) Call the tangent point of the A-excircle to side \(BC\) be \(D\). Connect \(AD\). Do so similarly for the other two sides. The intersection of these three segments is the Nagel point.

Note by Alan Yan
1 year, 1 month ago

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The question is easy. Firstly it is obvious that if triangle is equilateral then then the Nagel point and the incentre coincide. So we only need to prove the other side that if the Nagel point and the incentre coincide then the triangle is equilateral.

So my proof is as follows. Firstly denote the point where AB and AC touch the excircle as P and Q respectively. Observe that AP=AQ. Also observe that BD=BP,CD=CQ. Hence we obtain that AB+BD=AC+CD. Also by angle bisector theorem we have AB/BD=AC/CD. Solving we obtain that BD=DC. Substituting we get AB=AC. Same applies for some other excircle and so AB=AC=BC. Hence proved :) Shrihari B · 10 months, 4 weeks ago

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@Shrihari B Here is another solution:

In barycentric coordinates, the Nagel point is \((s-a : s-b: s-c)\) and the incenter is \((a:b:c)\). Thus, they concur if and only if \[\frac{s-a}{a} =\frac{s-b}{b} = \frac{s-c}{c} \implies a = b = c\] Alan Yan · 10 months, 4 weeks ago

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@Alan Yan Hmm ..but i don't know actually what barycentric coordinates are . But i would like if u would post some more geometry problems. Shrihari B · 10 months, 4 weeks ago

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@Shrihari B https://brilliant.org/discussions/thread/geometry-problem-2-2/ Alan Yan · 10 months, 4 weeks ago

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Can you define what the Nagel point is? Calvin Lin Staff · 1 year, 1 month ago

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@Calvin Lin I have defined it. Alan Yan · 1 year, 1 month ago

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