Find the Circumradius for a Triangle

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SUMMARY

The discussion focuses on calculating the circumradius (R) of an acute-angled triangle ABC with orthocenter H, where the products and sums of the cosines of the angles are given specific values. The first question was solved using the relationship AH = 2R.cosA, leading to the result of 3/14R. The second question, which seeks the value of R, is noted to have an answer of 3/2, although the user struggles to establish a connection between R and the provided information. The acute nature of the triangle ensures that all cosine values are positive, which is crucial for the calculations.

PREREQUISITES
  • Understanding of triangle properties, specifically orthocenters and circumradii.
  • Knowledge of trigonometric functions, particularly cosine values in relation to triangle angles.
  • Familiarity with geometric relationships involving angles and sides of triangles.
  • Ability to manipulate algebraic expressions involving trigonometric identities.
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  • Study the derivation of the circumradius formula for triangles, focusing on acute-angled triangles.
  • Learn about the properties of orthocenters and their significance in triangle geometry.
  • Explore trigonometric identities and their applications in solving geometric problems.
  • Investigate the relationship between the angles and sides of triangles using the Law of Cosines.
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Mathematicians, geometry enthusiasts, and students studying advanced triangle properties and trigonometric relationships will benefit from this discussion.

AGNuke
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Let ABC is an acute angled triangle with orthocentre H. D, E, F are feet of perpendicular from A, B, C on opposite sides. Let R is circumradius of ΔABC.

Given AH.BH.CH = 3 and (AH)2 + (BH)2 + (CH)2 = 7, answer the following
Q1.
\frac{\prod \cos A}{\sum \cos^{2}A}Q2. What is the value of R?

ANS 1. From properties of triangle, the distance of Orthocentre from a point A is given by AH = 2R.cosA. Using the values of cosines and from information in the question, I solved the first question to get the answer 3/14R.

Now I have no clue on how to approach Q2. I can't seem to find any relation between the value of R and information given to me. BTW, from what I know, the answer mentioned is 3/2.
 
Physics news on Phys.org
What does the answer to Q1 tell you?
Why is it important that the triangle is acute angled?
 
Acute angled triangle means all the cosines are greater than zero?
 

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