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Twenty Sutra for Success.....

Mathematics for Piyush is a Passion from his childhood he was so passionate about Mathematics used to play with Numbers draw figures and try to get sides distance one day I draw a AP SERIES Right Angle Triangle( thinking that the distance between the point of intersection of median & altitude at the base must be sum of rest sides that was in My Mind). And at last Piyush Succeed. This new Theorem proved with Four Proof(Trigonometry/Co-ordinates Geometry/Acute Theorem/Obtuse Theorem). Here are the Proofs: This Theorem applies in Two Conditions: The Triangle must be Right-Angled. Its Sides are in A.P. Series. Proof with Trigonometry Proof with Obtuse Triangle Theorem Proof with Acute Triangle Theorem Proof with Co-ordinates Geometry Four Proof ( TRIGONOMETRY/CO-ORDINATES/OBTUSE TRIANGLE/ACUTE TRIANGLE) (By PIYUSH GOEL) Theorem: In a Right-Angled Triangle with sides in A.P Theorem: In a Right-Angled Triangle with sides in A.P. Series, the distance between the point of intersection of median & altitude at the base is 1/10 Th the sum of other two sides. This Theorem applies in Two Conditions: The Triangle must be Right-Angled. Its Sides are in A.P. Series. Proof with Trigonometry tan α =AD/DC AD= DC tan α —————–1 tan2α = AD/DE AD= DE tan 2 α —————-2 DC tan α = DE tan 2α (DE+EC) tan α = DE tan 2α DE tan α + EC tan α = DE tan 2α DE tan α + EC tan α = 2 DE tan α / (1- tan^2 α) DE tan α – DE tan^3 α + EC tan α –EC tan^3 α = 2 DE tan α EC tan α –EC tan^3 α- DE tan^3 α = 2 DE tan α – DE tan α tan α (EC – EC tan^2 α – DE t an^2 α )= DE tan α DE tan^2 α – DE = EC tan^2 α – EC -DE ( tan^2 α + 1) = -EC (1 – tan^2 α ) DE (sin^2 α /cos^2 α + 1) = EC (1- sin^2 α /cos^2 α ) DE (sin^2 α + cos^2 α /cos^2 α ) = EC (cos^2 α – sin^2 α /cos^2 α ) DE (sin^2 α + cos^2 α ) = EC(cos^2 α -sin^2 α ) DE (sin^2 α + cos^2 α ) = EC (cos^2 α -sin^2 α) ……..where (sin^2 α + cos^2 α =1) & (cos^2 α -sin^2 α = cos 2 α ) DE= EC cos 2 α cos α =a/a+d & sin α = (a-d)/ (a +d) cos^2 α = a 2/ (a +b) 2 sin^2 α = (a-d) 2/ (a+ d) 2 DE= EC (cos^2 α – sin^2 α ) = EC ( a 2 / (a +b) 2 – (a-d) 2/ (a +d) 2 = EC ( a 2 – (a-d) 2/ (a +d) 2 = EC (a –a +d) (a+ a-d)/ (a+ d) 2 = EC (d) (2 a -d)/ (a+ d) 2 = (a +d)/2(d) (2 a -d)/ (a +d) 2 ————- where EC= (a +d)/2 = (d) (2 a -d)/2(a +d) = (d) (8 d -d)/2(4 d+d) ——————where a= 4 d (as per the Theorem) = 7 d 2 /2 (5 d) = 7 d /10 = (3 d+4 d)/10= (A B+AC)/10 Proof with Obtuse Triangle Theorem AC 2=EC 2 +A E 2 +2 C E. DE where EC = ( a +d) /2,A E=( a +d)/2 a 2 = (a +d/2)2 + (a+ d/2)2 + 2(a +d)/2 DE = (a +d/2) (a+d+2 DE) = (a +d/2) (a+d+2 DE) where a= 4 d 16 d 2 = (5 d/2) (5 d+2 DE) 32 d/5 = 5 d + 2 DE 32 d/5 – 5 d = 2 DE 32 d -25 d/5 = 2 DE DE =7 d /10 = (3 d+4 d)/10 = (A B+AC)/10 Proof with Acute Triangle Theorem A B 2= AC 2+BC 2 – 2 BC.DC (a-d) 2= a 2 + (a+ d) 2 -2(a+ d) (DE+EC) where A B= (a-d), AC=a, BC =( a +d) & EC= (a +d)/2 (a-d) 2 – (a +d)2 = a 2 -2(a +d)(DE+EC) (a- d –a-d) (a -d +a +d) = a 2 -2(a+ d) (2 DE+a+d)/2 2(-2 d) (2 a) = 2 a 2 -2(a +d) (2 DE+a+d) -8 ad – 2 a 2 = -2(a +d) (2 DE+a+d) -2 a (4 d +a) = -2(a +d) (2 DE+a+d) a (4 d + a) = (a +d)(2 DE+a+d) 4 d (4 d + 4 d) = (4 d+d) (2 DE+4 d+d) 4 d (8 d) = (5 d) (2 DE+5 d) 32 d 2/5 d = (2 DE+5 d) 32 d/5 = (2 DE+5 d) 32 d/5 – 5 d = 2 DE (32 d – 25 d)/5 = 2 DE DE = 7 d/10 = (3 d+4 d)/10 = (AB+AC)/10 Proof with Co-ordinates Geometry In Triangle A B C,point A,B&C,s co-ordinates are respectively (0,0) ,(a,0)&(0,b). Point D is middle point ,co-ordinates of Point Dis (a/2,b/2) Equation of BE is ……………… ( Two Points equation) Y –Y1 =(X – X1 )(Y2 –Y1)/(X2 – X1 ) Y – 0 =b-0/0-a(X – a) Y = -b/a(X) + b——————- (1) M 1 = -b/a For perpendicular M1 M2= -1 M2= -1/M1 So M2= a/b Equation of AC Y – 0 = a/b(X-0) Y=a/b(X) —————— (2) Put Y value in equation (1) a/b(X) + b/a(X) =b X (a 2+b 2/a b) = b X = a b 2/ (a 2 + b 2) To get Value of Y, put X value in equation (2) Y = a/b (a b 2/ (a 2+b 2) Y = a 2 b/ (a 2+b 2) Here we got co-ordinates of Point C – a b 2/ (a 2 + b 2), a 2 b/ (a 2+b 2) and co-ordinates of point D is (a/2, b/2) because d is midpoint. As per the A.P Series (z-d,z,z+ d) Hers a= z-d, b= z, c = z+ d (z +d) 2= (z-d) 2+ z 2 (z +d) 2- (z-d) 2 = z 2 (z +d +z-d)(z +d – z + d) = z 2 (2 z)(2 d) = z 2 4 z d = z 2 4 d = z Put value of a & b a b 2/ (a 2 + b 2), a 2 b/ (a 2+ b 2) & (a/2, b/2) a b 2/ ( a 2 + b 2) = 48 d/25 a 2 b/ (a 2+b 2) = 36 d /25 a/ 2=3 d/2 b/ 2 =4 d/2 CD 2= (48 d/25 -3 d/2)2-(3 6 d/25-4 d/2)2 = (96 d-75 d/50)2 + (72 d-100 d/50)2 = (21 d/50)2 + (-28 d/50)2 = (441 d 2/2500) + (784 d 2/2500) = (1225 d 2/2500) CD= 35 d/50 = 7 d/10 = 7 d/10 = (3 d+4 d)/10 = (A B+A E)/10 Copyrighted to PiyushGoel. (https://journals.pen2print.org/index.php/ijr/article/view/3743/3589)

Piyush Goel by profession he is Mech Engg but by passion he is Mirror Image Man, Cartoonist, Local Cricket Umpire,has completed 16 Books in Mirror Image written with different Objects like Pen, Needle, Mehandi Cone,Carbon Paper, Fabric Cone Liner,Iron Nail and with Wooden Pen on Magic Sheet. Piyush Goel writer of World First Hand write Needle Book Madhushala, here we are discussing Main Five books ShreeMad Bhagwad Gita, Madhushala, Panchtantra, Gitanjali and Piyush Vaani. 1.Mirror Imaged Shree Mad BhagvadGita (Pen)-Piyush Goel has a unique art, he can write words in mirror image.Piyush Goel can write words in mirror image in two languages Hindi anrd English.He has completed “World First Mirror Image Book Shreemad BhagvadGita”, all 18 chapters, 700 verses in two languages Hindi and English.Besides all of these he has completed “Shree Durga Sapt Satti” in Sanskrit Languages,Shree Sai Satcharitra in two languages Hindi and English,Sundar Kand (two times). 2. Needle Write Book “Madhushala”(Needle)-To read a book in front of a mirror ,reading then mirror very hard.People asked Piyush to read your mirror imaged books,mirror is necessary, he replied “Yes”.After a lot of thinking an idea stuck into his mind “Why should i not write a book with Needle.Then he started writing “Madhushala”,a book of Harbans Rai Bachchan,father of Amitabh Bachchan,Legend of Indian Cinema,now no need of Mirror.(Mirror image but no need of mirror). 3.Book “Gitanjali”written with Mehndi Cone(Mehdi Cone)-Piyush always think “DO SOMETHING NEW”,because of this thinking,he make a project and work on it.Whenever he has spare time and bring out the work at the end and this is the result of “Gitanjali”a book of Noble Literate Rabindra Nath Tagore,completed with the help of Mehndi Cone. 4. Book “Piyush Vani” with the help of Iron Nail (Iron Nail)-Imagination and Creativity brings new things to the world, Piyush started writing “Piyush Vani” one of his own published book, written on A-4 size Aluminium Sheet with the help of “Iron Nail”.(Mirror image but no need of mirror). 5. Book ‘Panchtantra”with the help of Carbon Paper (Carbon Paper)-Recently Piyush Goel completed “Vishnu Sharma”,s “Panchtantra” 5 tantras ,41 stories with the help of Carbon Paper,on one page both the words(mirror/non mirror) appeared in front and in the reverse.(Mirror image but no need of mirror).