ISSN: 2689-7636

Short communication
Open Access Peer-Reviewed

Kanjampatti P.O, Pollachi Via, Tamil Nadu 642003, India

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**Copyright Licence**

**MSC:** 08C99.

**PACS:**

Let ABC be the given triangle. On BC choose points D, E, F, and G. Join AD, AE, AF, and AG.

Let the sum of the interior angles of triangles ABD, ADE, AEF, AFG, and AGC be x, y, z, m, and n respectively Figure 1.

Assuming the angles at D, E, F and G as straight angles and adding we get that,

x + y = v + a where v is 180 degree (1)

y + z = v + b (2)

z + m = v = c (3)

m + n = v + d (4)

Let us formulate,

A = x – a

-y +v

B = y – b

v - z

C = v – z

m -c

D = n – d

v - m

xv -ya –yz +bv – vc +mz -mn +vd = 0

v[ b + d +x – c] +z[ m – y] –ya – mn = 0 (5)

(2) – (3) gives, b – c = y – m (6)

Applying (6) in the first factor of (5) we have,

v[ d +x + y - m] +z[ m – y] –ya – mn = 0

i.e v[ x +d] +z[ m – y] + y[ v – a ] –m [ v +n] = 0 (7)

If AEF is an isosceles triangle and if ED = FG, the triangles ADE and AFG are congruent/

Consequently we obtain that y = m. (8)

Applying y = m in equation (7) we get that,

v[ x +d] + y[ v – a - v - n] = 0

i.e v[ x +d] - y[ a + n] = 0

i.e v[ x +d] - = y[ a + n] (9)

Assuming y = m and (1) – (4) gives that x +d = n + a (10)

Applying (10) in (9) we obtain that y = v (11)

Since v is equal to 180 degrees by (1) to (4) we have that the sum of the interior angles of triangle ADE is equal to 180 degrees.

The recent probes and observations of NASA’s WMAP and ESA’s Planck spacecraft revealed that the geometry of our universe is flat with a slight margin of error of 0.4%. But there is no mathematical formulation and rigorous proof. The author has published eight different proofs [1-8]. Various theories for the end of our universe have been proposed namely, Big Freeze or Heat Death, Big Rip, Big Crunch. Now we have turned our focus on the cosmic topology.

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- S Kalimuthu (2020) On freedman equation and the shape of our universe. Ann Math Phys 30-31. Link: https://bit.ly/36HIs7I
- S Kalimuthu (2021) On Friedman equation, quadratic laws and the geometry of our universe. Int J Phys Res Appl 48-50. Link: https://bit.ly/3IAAjze

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