Question:
circles (concyclic, 3 pts given and do they form equilateral, right angled triangle)
circles (concyclic, 3 pts given and do they form equilateral, right angled triangle)
Updated On: Jun 23, 2024
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Solution and Explanation
To determine if three points are concyclic (lie on the same circle), we can use the concept of the circumcenter. The circumcenter is the center of the circle that passes through all three points.
If three points form an equilateral triangle, they are always concyclic. An equilateral triangle has all three sides of equal length and all three angles equal to 60 degrees.
If three points form a right-angled triangle, they may or may not be concyclic. A right-angled triangle has one angle measuring 90 degrees. However, for the three points to be concyclic, the right angle must be subtended by the diameter of the circle.
To check if the given three points form an equilateral triangle or a right-angled triangle, you need to provide the coordinates or describe the points in more detail.
If three points form an equilateral triangle, they are always concyclic. An equilateral triangle has all three sides of equal length and all three angles equal to 60 degrees.
If three points form a right-angled triangle, they may or may not be concyclic. A right-angled triangle has one angle measuring 90 degrees. However, for the three points to be concyclic, the right angle must be subtended by the diameter of the circle.
To check if the given three points form an equilateral triangle or a right-angled triangle, you need to provide the coordinates or describe the points in more detail.
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Concepts Used:
Three Dimensional Geometry
Mathematically, Geometry is one of the most important topics. The concepts of Geometry are derived w.r.t. the planes. So, Geometry is divided into three major categories based on its dimensions which are one-dimensional geometry, two-dimensional geometry, and three-dimensional geometry.
Direction Cosines and Direction Ratios of Line:
Consider a line L that is passing through the three-dimensional plane. Now, x,y and z are the axes of the plane and α,β, and γ are the three angles the line makes with these axes. These are commonly known as the direction angles of the plane. So, appropriately, we can say that cosα, cosβ, and cosγ are the direction cosines of the given line L.
