Enigma

Enigmes & Problèmes scientifiques

Euclide Geometry Problem

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

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

From Gogeometry Using : Additional points Similar triangles Equilateral triangle Define 0 middle of BC. O is the center of semicircle OD diameter BC Define a=OB=OD=OC => AB=AC=BC=2a Arc BD=arc DE=arc EC => ∠BOD=∠DOE=∠EOC=Π/3 OB=OD=a and ∠BOD=Π/3 =>BD=a, ΔBOD is equilateral and BD//AC In the same way, DE=a, ΔDOE […]

Gogeometry Problem 326

dans Euclide Geometry Problem étiqueté Equal arcs par Harvey Littleman

From Gogeometry Using : Pythagoras Proposition 13 Euclide Book II In ΔDMF : DM^2=DF^2+FM^2 (1) DM^2= (r-x/2)^2+x^2 In ΔAMD : AM^2=AD^2+DM^2 -2xAD (Proposition 13 Euclide Book II) r^2= r^2+ DM^2 -2xr (2) DM^2=2xr (1) and (2) 2xr=(r-x/2)^2+x^2 2xr=r^2-xr+x^2/4+x^2 3xr=r^2+5x^2/4 Dividing by x^2 (x not equal to 0) =>3r/x=5/4+(r/x)^2 Let y=r/x […]

Gogeometry Problem 336

dans Euclide Geometry Problem étiqueté a Common Tangent and a Square / Two equal circles par Harvey Littleman

From Gogeometry Using : Isometric transformation Additional points Pythagoras Similar triangles Tangents to a circle Define O middle of AB, r=ED AD^2+AO^2=DO^2 a^2+(a/2)^2=(a/2+r)^2 a^2+a^2/4=a^2/4+r^2+ar r^2+ar-a^2=0 ∆=a^2+4a^2=5a^2 r>0 => r=(a.sqr(5)-a )/2 Draw a square symetryc of ABCD by O =>D becomes D’, C becomes C’, E becomes E’ => C’D’BA is […]

Gogeometry Problem 458

dans Euclide Geometry Problem étiqueté Circular Sector / Internal Common Tangent / Measurement / Semicircle / square par Harvey Littleman