Three random points – Triangle & Circle

Introduction

When you drop three random points, all of you know that – if the third point is not on the same line with other two points, then we can draw a triangle connecting those three points.

And few of you well aware that a circle can be formed touching edges of the triangle all the time even if the points vary in location except that if all three points fall on the same line, in that case we might think only two points that circle can touch. However circle close to infinite size may give that small line on its perimeter of the circle.

At the sametime an inner circle can be drawn inside the triangle that can perfectly touch all the sides of the triangle. Even if all the three points fall on the same line, all individual points that constitutes the line that of size circle can be considered as inner circles that touches all the edges.

Properties of Triangle

Take an example of triangle . Here the angles are A, B & C. The sum of angles A+B+C will be always 180 degree.

Points for understanding

The table values for Sine, Cosine, Tangent are fit only for 90 degree triangle. In that triangle, the sum of the other two angles can only be 90 degree. So it is proportional. This yields ratio between adjacent side, opposite side and hypotenuse to arrive to some value.

So for normal triangle which doesn’t have an angle 90 degree this ratios will not fit. However it is true that when you drop perpendicular line from one edge to its opposite side, you will get two 90 degree triangle.

So the ratios Sine, Cosine, Tangent which are defined is still applicable. However when you take the whole triangle for consideration the formulas applicable are can be derived based on the fact of sine, cosine, tangent proportion and its ratio & relation between the other properties like R, S and so on.

S – Circum center
R – Distance between Circum Center and the edges.

Sine Formula

[To be Written]

Cosine Formula

[To be Written]

General Formulas in Trigonometry that can be fit & applied for any radical systems:

1) A + B + C = 180
….
[To be Written]

2D-Graph & Triangle

Point Lies On the Circle

Point lies on the circle will be . Here r is the radius of the circle. And is the angle [0 – 360 degree] with respect to X-Axis; Anti-clock wise; and from the mid point.

Transitive Angle Calculation

[To be Written]
….

Several Trigonometrical Equations

Several trigonometrical equation in which L.H.S is proven is equal to R.H.S
L.H.S – Left Hand Side
R.H.S – Right Hand Side
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