two isosceles triangles stuck together

the powerpoint is here

exploring how two isosceles triangles can be put together to create a triangle
or, how some triangles can be dissected into two isosceles triangles
exploring the angle relationships for the various options

isosceles triangle proofs

updated version

powerpoint is here

angle proofs

these are quite hard to solve but don't need circle theorems

the angles are constant








thanks to gogeometry
and the UKMT junior challenge
and Doug French, when he was at Hull University
for several of these ideas

g.m. in a right angled triangle

it is not too daunting to prove (or "provide a logical explanation") why the length of the altitude of a right angled triangle is the geometric mean of the two segments either side of the foot of the altitude (where it splits the hypotenuse)

this result in the Russian geometry texts of Kiselev (born 1852) is introduced after proportionality
(translated by Alexander Givental into English)
pdf version of the first book, 'planimetry'
and forms his proof of the 'pythagoras' theorem (in chapter 6, theorem 188)

it is simply established using two similar triangles

or you could use the intersecting chord theorem (if you are fortunate enough to know it) - usually established from similar triangles

how was this used as a method to create a square equal in area to a rectangle?

you could use pythagoras to establish the result
but if you used similarity, how could this result be used to prove the pythagoras theorem?

Thebault's theorem

Victor Thebault is credited with this theorem (1930)

create two equilateral triangles
on two adjacent sides of a square

join the three points as shown

prove that this (red dotty) triangle must be an equilateral triangle











you can do this by considering:
angles and establishing congruence;

or by other methods

what happens if the two triangles are drawn inside rather than outside the square?

perps

draw two lines, AB and AC
with any point, D, 'between' them

drop perpendiculars from D to AB (at F) and from D to AC (at E)

join E to F (the feet of these perpendiculars)

then create a perpendicular from A to EF (meeting EF at G)


possibly set this up (e.g. with GeoGebra)

establish, by measuring and dragging D around
that angle GAE = angle FAD

try to prove it

can students find another angle the same as these?

isometric angles

when tackling angle work there seems to be merit in restricting the angles to those found on an isometric grid

with an isometric grid the basic shape of an equilateral triangle can be used to determine other angles

see potentially earlier work on 7 pins

later work can then focus on deciding upon and establishing (i.e. justifying) what various angles are - from a restricted set, to work towards establishing the interior angle sum for various polygons

the 'other' angle to make 360 degrees might also be of interest
'explementary' seems to be a (rarely used) term for this

nonagon appreciation

a nonagon has several properties involving some nice simple angles

a ppt is here

show that a + b = c