Enigma

Enigmes & Problèmes scientifiques

Harvey Littleman

From Gogeometry Using Additional points Pythagoras congruent triangles Similar triangles Equilateral triangle Let b=AD=DC and d=BCΔACB and ΔBCD are similar since ∠BAC=∠DBC=x and ∠BCD=∠BCAThen AC/BC=CB/CD=AB/BD => 2b/d=d/b => d=b√2 Define C’ such as DC’ ⊥ AC and DC’=b∠BDC=∠ADC – π/4= π – π/4 = 3 π/4∠BDC’=∠BDA + ∠ADC’ = π/4 […]

Gogeometry Problem 001

dans Euclide Geometry Problem / Géométrie Euclidienne étiqueté 45 Degrees / Angles / Congruence / Midpoint / Triangle par Harvey Littleman

From Gogeometry Using Additional points Inscribed angles in a circle congruent triangles Concyclic points Solution of a previous problem : 496 CK⊥ KH and CE⊥ EH =>CKEHJ are concyclic=> ∠EKC= 3Π/2= ∠EKJ + ∠JKC = ∠EKJ + ∠JEC = ∠EKJ + Π/2=> ∠EKJ= Π/2K and I on GJ, IE⊥ GJ […]

Gogeometry Problem 1291

dans Euclide Geometry Problem / Géométrie Euclidienne étiqueté Collinear Vertices / Midpoint / Three Squares par Harvey Littleman

From Gogeometry Using : Additional points Similar triangles •ΔABC is isosceles and right in B => BA=BC and ∠BAC=∠BCA=Π/4 • Define G in AC such as AB=AG => ΔBAG is isosceles in A •Define H intersection of AD and BG •∠BAH=∠GAH=Π/4/2=Π/8 and HB=HG and ∠AHB=∠AHG=Π/2 •and ∠ABH=∠AGH and ∠ABH+ ∠BAH […]

Gogeometry Problem 975

dans Euclide Geometry Problem / Géométrie Euclidienne étiqueté 45 Degrees / Angle Bisector / Hypotenuse / Isosceles Right Triangle par Harvey Littleman

From Gogeometry Using : Additional points Tangents to a circle •(1) AF=ra=r+EB+BF=r+a2 •(2) AG=ra=r+DC+CG=r+a1 •(1-2) a1=a2 •(1+2) ra=r+a1=r+a2 => a1=a2=ra-r •Define X as the intersection of BC and circle ra •Define Y as the intersection of BC and circle r •CD and CY tangents to circle r => CD=CY •BE […]

Gogeometry Problem 207

dans Euclide Geometry Problem / Géométrie Euclidienne étiqueté Exradius relative to the hypotenuse par Harvey Littleman

From Gogeometry Using : Isometric transformation Inscribed angles in a circle Equilateral triangle Tangents to a circle Define R radius of circle O and r radius of circle Q By construction and symmetry A, O, Q, D are collinear DE=DF, AE=AF AD angle bisector of ∠BAC D symmetric of A […]

Gogeometry Problem 1362

dans Euclide Geometry Problem / Géométrie Euclidienne étiqueté Equilateral Triangle Inscribed in a Circle / Lunula / Ratio of Areas / Sketch / Tangent par Harvey Littleman

Using : Additional points Pythagoras Similar triangles Thales theorem Tangents to a circle

Problem H003

dans Euclide Geometry Problem / Géométrie Euclidienne étiqueté circle par Harvey Littleman

From Gogeometry Using : Pythagoras Inradius in right triangle Define r as the inradius of ΔABC in a right triangle, r=(AB+BC-AC)/2 AB=AC => r=(2AB-AC)/2=AB-AC/2 BI=r.sqrt(2)=sqrt(2)(AB-AC/2)=sqrt(2).AB-AC/sqrt(2) AC=sqrt(2).AB => BI=AC-AB Therefore AC=AB+BI

Problem 976

dans Euclide Geometry Problem étiqueté 45 Degrees / Angle Bisector / Hypotenuse / Incenter / Isosceles Right Triangle par Harvey Littleman

ΔABC right triangle in C Define r its inradius, a=BC,b=AC and c=A PCRO is square => PC=CR=r AC et AB tangents to the circle=>AP=AQ, BQ=BR and CP=CR b=AC=AP+PC, a=BC=BR+RC AC+BC=b+a=AP+r+BR+r=2r+AP+BR AP=AQ, BR=BQ => AC+BC=b+a=2r+AQ+BQ=2r+AB=2r+c a+b=2r+c Therefore r=(a+b-c)/2 Define p as the semi-perimeter of the triangle, p=(a+b+c)/2 Therefore r =p-c

Inradius in right triangle

dans Euclide Geometry Principles par Harvey Littleman

define a CB, b=AC and c=AB Define S as the area of the triangle S=r.a/2 + r.b/2 + r.c/2 Therefore S=r(a+b+c)/2 and r=2S/(a+b+c)

Inscribed circle of a triangle

dans Euclide Geometry Principles par Harvey Littleman

From Gogeometry Define a=AB=BC=AC DC and DB tangents to Circle O1 => DG=DE=bAD and AB tangents to Circle O1 => AG=AJ=dDB and BA tangents to Circle O1 => BE=BJ=c DB and BC tangents to Circle O2 => BF=BK=eDC and BC tangents to Circle O2 => CI=CK=f a=d+c=e+fDF=b+c+e, DI=b+d+a+fDC and DB […]

Gogeometry Problem 1372

dans Euclide Geometry Problem étiqueté Altitude / Equilateral Triangle / Exradius / Exterior Cevian / Inradius par Harvey Littleman