Exam (elaborations)
Identities and Inequalities in Geometry(2)
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Identities and Inequalities in Geometry(2)
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Useful Identities and Inequalities in Geometry 1Contributors in Posting Order
Samer Seraj (BigSams)2
Andrew Kirk (WakeUp)
Réda Afare (Thalesmaster)3
Luis González (luisgeometra)
Constantin Mateescu (Mateescu Constantin)
Typesetting and Editing
Ercole Suppa 4 (Leon)
Samer Seraj (BigSams)
June 29, 2011
1 Regular Notation for a 4ABC
• Let AB = c, BC = a, CA = b be the sides of 4ABC.
• Let A = m (∠BAC), B = m (∠ABC), C = m (∠BCA) be measures of the angles of 4ABC.
• Let ∆ be the area of 4ABC.
• Let P be any point in 4ABC. Let the cevians through P and A, B, C intersect a, b, c at Pa , Pb ,
Pc respectively.
• Let arbitrary cevians issued from A, B, C be d, e, f respectively.
• Let the semiperimeter, inradius, and circumradius be s, r, R respectively.
• Let the heights issued from A, B, C be ha , hb , hc respectively, which meet at the orthocenter H.
• Let the feet of the perpendiculars from H to BC, CA, AB be Ha , Hb , Hc respectively.
• Let the medians issued from A, B, C be ma , mb , mc respectively, which meet at the centroid G.
• Let the midpoints of A, B, C be Ma , Mb , Mc respectively.
• Let the internal angle bisectors issued from A, B, C be `a , `b , `c respectively, which meet at the
incenter I, and intersect their corresponding opposite sides at La , Lb , Lc respectively.
• Let the feet of the perpendiculars from I to BC, CA, AB be Ia , Ib , Ic respectively.
• Let the centers of the excircles tangent to BC, CA, AB be Oa , Ob , Oc respectively, and the excircles
be tangent to BC, CA, AB at Ea , Eb , Ec .
• Let the radii of the excircles tangent to BC, CA, AB be ra , rb , rc respectively.
• Let N be the Nagel Point, and let Γ be the Gergonne Point.
Important: CVH denotes that Cyclic Versions Hold
To refer to this document, especially on Mathlinks, call it IIG (Identities and Inequalities in Geometry)
1 Theoriginal thread: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=50&t=412623/
2 Email: samer seraj@hotmail.com
3 Email: master 8701@hotmail.com
4 Email: ercolesuppa@gmail.com, Web: http://www.esuppa.it/
1
, 2 Trigonometric Identities for General Angles θ, φ
Note: Any identity that is included for the regular functions, but not for the reciprocal functions, can be
converted to the latter by replacing the regulars by the reciprocals and some simple cross multiplication
and fraction manipulation.
1. Reciprocal Identities
1
(a) sin θ =
csc θ
1
(b) cos θ =
sec θ
1
(c) tan θ =
cot θ
2. Pythagorean Identities
(a) sin2 θ + cos2 θ = 1
(b) tan2 θ + 1 = sec2 θ
(c) 1 + cot2 θ = csc2 θ
3. Quotient Identities
sin θ
(a) tan θ =
cos θ
cos θ
(b) cot θ =
sin θ
4. Co-function Transformation Identities
(a) sin(θ + 90◦ ) = cos θ
(b) tan(θ + 90◦ ) = − cot θ
(c) sec(θ + 90◦ ) = − csc θ
5. Reflection Identities
(a) sin(180◦ − θ) = sin θ
(b) cos(180◦ − θ) = cos θ
(c) tan(180◦ − θ) = − tan θ
6. Period Identities (best constants obviously), n ∈ Z
(a) sin(θ + 360◦ n) = sin θ
(b) cos(θ + 360◦ n) = cos θ
(c) tan(θ + 180◦ n) = tan θ
2
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