Client authentication (certificate + private key) using WinInet - ssl

Question

I'm trying to communicate in https with a server using WinInet (from the Win32 API).
Here is a very minimalist code :
HINTERNET ses = InternetOpenA("test",INTERNET_OPEN_TYPE_DIRECT,NULL,NULL,0) ;
HINTERNET con = InternetConnectA(ses,"stackoverflow.com",INTERNET_DEFAULT_HTTPS_PORT,NULL,NULL,INTERNET_SERVICE_HTTP,0,NULL) ;
HINTERNET req = HttpOpenRequestA(con,"GET",NULL,NULL,NULL,NULL,INTERNET_FLAG_SECURE,NULL) ;
HttpSendRequestA(req,NULL,0,NULL,0) ;
DWORD read ;
char str[3000] ;
InternetReadFile(req,reinterpret_cast<void*>(str),sizeof(str)-1,&read);
str[read] = 0 ;
cout << &str[0] << endl ;
As long as I communicate with a "classic" https server, like stackoverflow.com, everything goes well. The problem is when I try to communicate with a server that requests an authentication of the client.
I have 3 .pem files : a certificate and a private key for my client, and a root certificate that authenticates my client certificate (i.e. a certificate chain of length 2).
For information, I can connect my server using this cULR command line :
curl https://my.server --cert Client_cert.pem --key Client_key.pem --cacert Root_cert.pem
This is the proof that it's possible!
Reading the WinInet documentation, I found a page named "Handling Authentication", but it's all about username:password, and there's nothing about certificate.
I found out that I have to use the Crypt32 library : I create a certificate context with CertCreateCertificateContext (using binary data from client_cert.pem) and then pass it using InternetSetOptionA. But then, HttpSendRequestA fails with an error 12157...
I must admit that I would be glad to find a good tutorial or some code sample ! By the way, I don't have a piece of clue about how to insert my private key into that stuff...
Thanks in advance !