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Angle between 90 and 135 degrees.

by Rob
(Seattle)











































In triangle ABC, the altitude from B is tangent to the circumcircle of ABC. Prove that
the largest angle of the triangle is between 90 and 135. If the altitudes from both B and
from C are tangent to the circumcircle, then what are the angles of the triangle?

I honestly have no idea!!

Comments for Angle between 90 and 135 degrees.

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Aug 09, 2011
Angle between 90 and 135 degrees
by: Staff

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Part II



Calculating the maximum possible value of ∠A.

If point A is moved around the circumcircle away from point B, ∠A becomes larger and larger.



(1) Click the following link to VIEW a diagram of how the triangle looks when point A and point B are moved farther apart; or (2A) highlight and copy the link, then (2B) paste the link into your browser Address bar & press enter:

Use the Backspace key to return to this page:


http://www.solving-math-problems.com/images/circumcircle-triangle-03.png


As point A is moved away from point B along the circumcircle, ∠A will continue to increase until point A is located at the bottom of the circumcircle. At this point the points A and C would occupy the same position and the triangle would cease to exist.

However, the three angles in the RIGHT TRIANGLE formed by points A, B, and the side “h” triangle would be 90º, 45º, and 45º. This means ∠A = 135º. ∠A cannot actually reach 135º, but it can approach it.

Therefore,

∠A must always be larger than 90º, but less than 135º.

90º < ∠A < 135º



Thanks for writing.

Staff
www.solving-math-problems.com


Aug 09, 2011
Angle between 90 and 135 degrees
by: Staff

Part I

The question:

by Rob
(Seattle)

(1) In triangle ABC, the altitude from B is tangent to the circumcircle of ABC. Prove that
the largest angle of the triangle is between 90 and 135.

(2) If the altitudes from both B and from C are tangent to the circumcircle, then what are the angles of the triangle?

I honestly have no idea!!

The answer:

(1) In triangle ABC, the altitude from B is tangent to the circumcircle of ABC. Prove that
the largest angle of the triangle is between 90 and 135.

This is not a formal proof, but it will show you how to understand the lower and upper bounds for the largest angle of the triangle.



Diagram of the Problem:

(1) Click the following link to VIEW a diagram of the problem; or (2A) highlight and copy the link, then (2B) paste the link into your browser Address bar & press enter:

Use the Backspace key to return to this page:


http://www.solving-math-problems.com/images/circumcircle-triangle-01.png



The largest angle of the triangle is ∠A


Calculating the lower boundary of ∠A.


The vertices A, B, and C must all lie on the circumcircle.

As point A is moved closer to point B on the circumcircle, ∠A becomes smaller and ∠B becomes larger. The closer point A is to point B, the closer both∠A and ∠B are to 90º.


(Points “A” and “C” must be moved together so that the base AC always forms a right angle with altitude h. The altitude “h” must remain tangent to the circumcircle.)


(1) Click the following link to VIEW a diagram of how the triangle looks when point A and point B are closer together; or (2A) highlight and copy the link, then (2B) paste the link into your browser Address bar & press enter:

Use the Backspace key to return to this page:


http://www.solving-math-problems.com/images/circumcircle-triangle-02.png




Point A can never be positioned so that A and B occupy the same position on the circumcircle or the triangle would cease to exist.

If point A moves around the circumcircle beyond point B, ∠A becomes larger and larger.

∠A must always be larger than 90º. ∠A > 90º. This is the lower limit for the largest angle in the triangle.
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