Bước 1: Chứng minh △MNK=△ENKtriangle cap M cap N cap K equals triangle cap E cap N cap K
△???=△???
 
Xét △MNKtriangle cap M cap N cap K
△???
và △ENKtriangle cap E cap N cap K
△???
có: 
MN=NEcap M cap N equals cap N cap E
??=??
(giả thiết).
MNK̂=ENK̂modified cap M cap N cap K with hat above equals modified cap E cap N cap K with hat above
???=???
(do NKcap N cap K
??
là tia phân giác của MNP̂modified cap M cap N cap P with hat above
???
).
NKcap N cap K
??
là cạnh chung. 

Vậy △MNK=△ENKtriangle cap M cap N cap K equals triangle cap E cap N cap K
△???=△???
theo trường hợp cạnh - góc - cạnh (c.g.c). 

Bước 2: Chứng minh IE⟂NPcap I cap E ⟂ cap N cap P
??⟂??
 
Vì Icap I
?
là giao điểm của NKcap N cap K
??
và MPcap M cap P
??
, mà NKcap N cap K
??
là tia phân giác của MNP̂modified cap M cap N cap P with hat above
???
, nên Icap I
?
cũng nằm trên tia phân giác của góc Ncap N
?
.
Xét △MNItriangle cap M cap N cap I
△???
và △ENItriangle cap E cap N cap I
△???
có: 
MN=NEcap M cap N equals cap N cap E
??=??
(giả thiết).
MNÎ=ENÎmodified cap M cap N cap I with hat above equals modified cap E cap N cap I with hat above
???=???
(do NIcap N cap I
??
là phân giác).
NIcap N cap I
??
là cạnh chung. 

⇒△MNI=△ENIimplies triangle cap M cap N cap I equals triangle cap E cap N cap I
⇒△???=△???
(c.g.c).
⇒NMÎ=NEÎimplies modified cap N cap M cap I with hat above equals modified cap N cap E cap I with hat above
⇒???=???
(hai góc tương ứng).
Mà NMÎ=90∘modified cap N cap M cap I with hat above equals 90 raised to the composed with power
???=90∘
(do △MNPtriangle cap M cap N cap P
△???
vuông tại Mcap M
?
), nên NEÎ=90∘modified cap N cap E cap I with hat above equals 90 raised to the composed with power
???=90∘
.
Vậy IE⟂NPcap I cap E ⟂ cap N cap P
??⟂??
. 

Bước 3: Chứng minh MNÎ=QEP̂modified cap M cap N cap I with hat above equals modified cap Q cap E cap P with hat above
???=???
 
Vì EF∥MPcap E cap F is parallel to cap M cap P
??∥??
và NIcap N cap I
??
cắt chúng, nên EFÎ=MIN̂modified cap E cap F cap I with hat above equals modified cap M cap I cap N with hat above
???=???
(hai góc so le trong).
Mặt khác, từ △MNI=△ENItriangle cap M cap N cap I equals triangle cap E cap N cap I
△???=△???
, ta có MIN̂=EIN̂modified cap M cap I cap N with hat above equals modified cap E cap I cap N with hat above
???=???
.
Suy ra EFÎ=EIF̂modified cap E cap F cap I with hat above equals modified cap E cap I cap F with hat above
???=???
(vì F,Icap F comma cap I
?,?
cùng nằm trên đường thẳng NIcap N cap I
??
), dẫn đến △IEFtriangle cap I cap E cap F
△???
cân tại Ecap E
?
. Do đó EF=EIcap E cap F equals cap E cap I
??=??
.
Theo giả thiết IQ=FEcap I cap Q equals cap F cap E
??=??
, nên IQ=EIcap I cap Q equals cap E cap I
??=??
. Tam giác IEQcap I cap E cap Q
???
cân tại Icap I
?
.
Đặt MNÎ=α⇒ENÎ=αmodified cap M cap N cap I with hat above equals alpha implies modified cap E cap N cap I with hat above equals alpha
???=?⇒???=?
. Trong tam giác vuông MNIcap M cap N cap I
???
, ta có MIN̂=90∘−αmodified cap M cap I cap N with hat above equals 90 raised to the composed with power minus alpha
???=90∘−?
.
Vì M,I,Pcap M comma cap I comma cap P
?,?,?
thẳng hàng, góc bẹt MIP̂=180∘modified cap M cap I cap P with hat above equals 180 raised to the composed with power
???=180∘
. Ta có:
EIP̂=180∘−(MIN̂+EIN̂)=180∘−2(90∘−α)=2αmodified cap E cap I cap P with hat above equals 180 raised to the composed with power minus open paren modified cap M cap I cap N with hat above plus modified cap E cap I cap N with hat above close paren equals 180 raised to the composed with power minus 2 open paren 90 raised to the composed with power minus alpha close paren equals 2 alpha
???=180∘−(???+???)=180∘−2(90∘−?)=2?

Trong tam giác cân IEQcap I cap E cap Q
???
( IE=IQcap I cap E equals cap I cap Q
??=??
):
IEQ̂=180∘−EIP̂2=180∘−2α2=90∘−αmodified cap I cap E cap Q with hat above equals the fraction with numerator 180 raised to the composed with power minus modified cap E cap I cap P with hat above and denominator 2 end-fraction equals the fraction with numerator 180 raised to the composed with power minus 2 alpha and denominator 2 end-fraction equals 90 raised to the composed with power minus alpha
???=180∘−???2=180∘−2?2=90∘−?

Vì IE⟂NPcap I cap E ⟂ cap N cap P
??⟂??
, nên IEP̂=90∘modified cap I cap E cap P with hat above equals 90 raised to the composed with power
???=90∘
. Ta có:
QEP̂=IEP̂−IEQ̂=90∘−(90∘−α)=αmodified cap Q cap E cap P with hat above equals modified cap I cap E cap P with hat above minus modified cap I cap E cap Q with hat above equals 90 raised to the composed with power minus open paren 90 raised to the composed with power minus alpha close paren equals alpha
???=???−???=90∘−(90∘−?)=?

Vậy MNÎ=QEP̂modified cap M cap N cap I with hat above equals modified cap Q cap E cap P with hat above
???=???
(cùng bằng αalpha
?
). 

Đáp án: 
a) △MNK=△ENKtriangle cap M cap N cap K equals triangle cap E cap N cap K
△???=△???
(c.g.c).
b) IE⟂NPcap I cap E ⟂ cap N cap P
??⟂??
do NEÎ=NMÎ=90∘modified cap N cap E cap I with hat above equals modified cap N cap M cap I with hat above equals 90 raised to the composed with power
???=???=90∘
.
c) MNÎ=QEP̂modified cap M cap N cap I with hat above equals modified cap Q cap E cap P with hat above
???=???
.