All Questions
Tagged with triangles proof-writing
78
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How must I find the third vertex of an equilateral / right isosceles triangle given the coordinates of 2 vertices?
This question has been asked in different ways at different points of time in the Math SE, but I'm trying to look for proof-wise simplicity here.
If the ends of hypotenuse of a right isosceles ...
- 23
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4
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How to show that given two acute angles, the sine ratio of the greater angle is greater than the sine ratio of lesser angle?
How do I show that for angles $\theta ,\psi \in [0^°,90^°$], if $\theta > \psi$, then $\sin \theta > \sin \psi$?
I thought of proving this by creating two right triangles with the same ...
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Given points $A$, $B$, $C$ and $D$ lying on a circle and lines $BD$ and $AC$ intersecting at $F$, prove lines are parallel
The diagram shows the points $A$, $B$, $C$ and $D$ lying on a circle. $AC$ and $BD$ intersect at point $F$. $EG$ is tangent to the circle at point $C$. $AD$ is produced to meet the tangent at point $E$...
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Proving that no tile can fill both squares and equilateral triangles
Cut up a square into a finite number of identical tiles.
Here is one possibility:
How do I prove that the tiles could never be rearranged to form an equilateral triangle (with filled interior and no ...
- 791
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1
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65
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Maximize product of two sines given sum of angles [closed]
Say we have an angle $c$. Prove that for two angles $a$ and $b$ such that $a + b = c$,$$\sin{a}\sin{b}$$ is maximized when
$$a = b = \frac{c}{2}$$
- 119
5
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2
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128
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How to prove that the $\angle CED$ of the triangle below is equal to $\frac{1}{2} \angle\alpha\;?$
Here is the whole problem:
In the triangle $ABC$, it is known that $AC > AB$, and the angle at the vertex $A$ is equal to $\alpha$. On the side $AC$, point $M$ is marked so that $AB=MC$. Point $E$ ...
- 570
2
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76
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How to prove that given polygonal chain of line segments is greater than a given straight line?
The problem is from my textbook, the topic is "A midline of a triangle":
Given a triangle $ABC$ and points $D$ and $E$ such that $∠ADB =∠BEC = 90°$. Prove that $DE ≤ \frac{1}{2}P \triangle ...
- 570
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A didactic proof of the Pythagorean Theorem? [duplicate]
Does the following provide a didactically sound approach to the Pythagorean Theorem? We first consider the hypotenuse of a right isosceles triangle and then we extend the idea to a general right ...
- 2,040
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1
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Orthocenter of a triangle collinear with two points in the circumcircle.
This difficult elementary geometry problem was proposed by @Nyafh54 and receiving no answer was deleted twice despite several upvotes. We republish it here mainly for the information of the O.P. who ...
- 30.3k
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How to define a triangle's side for replacing with the condition on perimeter?
Suppose we have a triangle $\Delta ABC$ and $O$ is an inner point of triangle. It is required to determine which side of the triangle should be replaced on two line segments so that the resulting ...
- 1,231
2
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1
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304
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Difficult geometry Olympiad symmedian problem
I have been training for math olympiads for some time now. I came across this geometry problem from the Italian math book "Giochi Matematici Russi" by Boris A. Kordemsky:
Let ABC be an ...
user1087131
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1
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Similarity of triangles, theory 4, proof
On most of the internet sides I have read just 3 triangle similarity theorems, but I found out, there is also a 4: "Two triangles are similar if the lengths of two corresponding sides are ...
- 79
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Dividing a triangle into 4 different triangles - proof
I apologize for the rough and wobbly diagram but that was all I could achieve on Jamboard!
ABC is a scalene triangle and so is QPR.
The question is: "Prove that the perpendiculars from A to BC, B ...
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Why is $\frac{CD}{BD}=\frac{AC}{AB}$? [closed]
Here, $I$ is the in-center. According to my book,
$$\frac{CD}{BD}=\frac{AC}{AB}\tag{1}$$
My book didn't provide any justification for this claim. How do I prove $(1)$?
- 3,186
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Why the middle point of equilateral triangle is beneath its vertex?
I thought about this question when I thought why median and altitude of an equilateral triangle are the same.
And it seems to me it is all because the vertex is directly above the midpoint.
Though I ...
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